and Hecke triangle groups and OCR Output

cocompact but cofinite arithmetic groups, like Picard group PSL(2, Z[i])

included as a reference. The algorithm can be generalized to other non

spectrum of the Laplacian be simple. Examples of the numerical data are

Ramanujan—Peterss0n, Sato—Tate, and the conjecture that the discrete

tations confirm some important conjectures from number theory, namely

essentially on the theory of Hecke operators. The results of the compu

placian and of the Hecke operators for the modular group. It relies

ber of eigenvalues and eigenfunctions (Maaii wave forms) of the La

A new method is described to compute with high accuracy a large num

Abstract

Luruper Chaussee 149, 22761 Hamburg, Germany

Il. Institut fiir Theoretische Physik, Universitat Hamburg

Gunther Steil

the Hecke operators for PSL(2,Z)

Eigenvalues of the Laplacian and of

March 1994

ISSN 0418-9833DESY 94-028

termined as functions of R from the nonlinear system that arises from writing OCR Output

This fulfils eq. (1) and part of the invariance condition The c,,’s are de

tions with the coefficients expressed as polynomials in the prime coefficients cp.

The algorithm takes as starting point the Fourier expansion of the eigenfunc

simply as eigenvalues and to those of the Hecke operators as coefficients.

facilitate language we occasionally shall refer to the eigenvalues of the Laplacian

coefficients. Coefficients with non-prime index are polynomials in the c,,’s. To

eigenvalues Cp of the Hecke operators enter the Fourier expansion (20) of f (z) as

for details and the definition of the Hecke operators T, see section 2 below. The

(4)A = 3176; + 6;).The Laplacian is given by

f(*yz) = f(z), 7 E PSL(2,Z), Imz > O. (3)

subject to the automorhy condition

Tp f(z) = ¢pf(Z), P prime, (2)

(1)Aft?) = »\f(Z) = (i + R2) f(Z)»

simultaneously eigenvalues and square—summable eigenfunctions of

the Laplacian and of the Hecke operators for the modular group. We compute

curacy a large number of eigenvalues and eigenfunctions (MaaB wave forms) of

In this paper we present a numerical method for computing with high ac

can be found in [15, 37].

modular group are [6, 7, 8, 10, 13, 14, 16, 17, 18, 19, 20, 34, 36, 37, 42]; reviews

The relevant references concerning the computational work in the case of the

business by the discourse of quantum chaos.

techniques towards hard mathematics. Lately physicists got involved into this

more and more interest, indicating as well a creeping motion of new ”technical”

idea of getting numerical insight into this and related (dicrete) spectra attracted

In line with the increasing power of computers, in the past some 15 years the

known about its discrete part whose existence is not even a trivial fact [9, 23, 31].

spectrum can be described rather explicitly in terms of the Eisenstein series, less is

of the Selberg trace formula, cf. [15, 21, 35]. While the continuous part of the

to be one of the most interesting in modern number theory and the formalism

The spectrum of the hyperbolic Laplacian for the modular group P SL(2, Z) seems

1 Introduction and summary

Icp] s 2. (5) OCR Output

The Ramanujan—Petersson conjecture states

cients cp agree with the predictions of a theorem of Sarnak [32].

Keeping p fixed, it is seen (fig. 3) that the distribution of the Fourier coeffi

2). (See also [17].)

which include up to 30000 coefficients. They clearly confirm the conjecture (fig.

its validity we prepared extensive lists of c,,’s for a couple of Maass wave forms

be distributed according to Wigner’s semicircle law. To get numerical hints on

The Sato—Tate conjecture asserts that the c,,’s of any eigenfunction should

[4,

frequently (in the language of quantum chaos the spectrum is Poissonian, see

1). The spectrum is simple in the whole range, but nearly degeneracies occur

have been found, the largest of which lies above the 1.3 >< 106th eigenvalue (table

1). Additionally, a couple of large eigenvalues around R = 1000, 2000, and 4000

was checked by comparison of the results with a generalized Weyl’s law (cf. figure

include about 10000 eigenvalues (tables 4, 5 show examples). The completeness

Our computations cover the ranges 0 < R < 350 and 500 < R < 510 which

finite element approximations of eigenfunctions.

ficients/eigenfunctions in two recent papers [17, 19]. Huntebrinker [20] presents

could be useful in this respect [36]. Apart from this, only Hejhal considers coef

priate Fourier coefficients. Stark was the first to recognize that equation (2)

The determination of eigenfunctions essentially consists in finding the appro

developed the present method.

in [8, 34, 37]; the first two papers consider only odd eigenfunctions, the latter

have been known). More than one thousand eigenvalues were respectively found

[16] who was the first to go beyond R z 25 (so far only 20 correct eigenvalues

been computed by Cartier and Haas [7, 13]. Major progress is due to Hejhal

is the usual parameter) as solutions of eqs. (1) and The first eigenvalues have

Most of the cited papers deal with the determination of the eigenvalues R (this

eigenvalues recently. Results will be reported in future paper.

to a 3-dimensional system [22]. For the latter, we have been computing the first

the modular group [11], and for the Picard group PSL(2, Z[i]) which corresponds

the case for the subgroups (so—called Hecke triangle groups) 1`(\/E), m = 2, 3, of

other systems for which Hecke operators with similar properties exist. This is

remaining conditions (3) are fulfilled. This proceeding can also be applied to

down explicitely eq. (2) for the respective p. The eigenvalues are found iff the

orthogonal to the real axis. OCR Output

bolic geometry in two dimensions. Geodesics are semicircles and straight lines

H has constant negative Gaussian curvature K : -1 and is a model for hyper—

dz = yddrrdy. (9)

(8)A = $(8; + @3),

are

The (hyperbolic) Laplace operator and the volume element associated with ds?

(132 : y—2 (dm2 +dy2

be the Poincaré upper half plane which is equipped with the Riemannian metric

'H={z:m+iy€C|m€lf{,y>0} (6)

Let

Terras’ book [39].

presentation and many references to the original literature can be found e.g. in

necessary to construct the algorithm in the following section. A more detailed

In this section we shortly collect well—known basic notions and facts which are

2 Preliminaries

detailed informations consult [3, 4, 5, 33].

on arithmetic groups the notion of arithmetic chaos was recently introduced. For

structure of the modular group. For this and related systems which are based

matrix theory [29]. The origin of this exceptional behaviour lies in the arithmetic

invariant) chaotic systems exhibit GOE statistics as described by the random

of a sequence of equidistributed random numbers, whereas most (time-reversal

gies the spectrum shows Poissonian fluctuations, i.e. iiuctuations similar to those

spectrum look more like those of an integrable system. At reasonable high ener·

Nevertheless, it turned out [3, 4, 34, 37] that the statistical properties of the

main. The classical counterpart of this system has a. chaotic geodesic flow [2, 28].

energy levels of a quantum mechanical particle sliding freely on the modular do

for general informations) where the eigenvalues of the Laplacian correspond to the

The results are also of interest in the context of quantum chaos (see e.g. [12]

cients.

'We have not found any violation of this bound among more than 300000 coeffi

spectrum, and which is supposed but not proven to be simple. OCR Output

It contains the discrete spectrum of A which is embedded in the continuous

We are concerned with the third part Cusp of the spectral decomposition.

eigenvalue A0 = 0 which exists since .7: has finite hyperbolic area.

Constant contains the constant eigenfunctions associated with the trivial

functions in 1:.

spectrum of A which extents from i to oo. Functions in Eisenstein are even

Eisenstein is spanned by the Eisenstein series and contains the continuous

L2(l"\H) : Eisenstein EB Constant GB Cusp.sum [23, 31]

of them in the following. The spectrum of A splits L2(l`\H) into the orthogonal

fore has a unique self-adjoint extension We will not distinguish between both

(f, f) = fj, f(z)g(z) dz < oo. A is essentially self-adjoint on L2(1"\H) and thereLet L2(l`\H) be the Hilbert space of square summable automorphic functions

z E H.

automorphic functions on H which have to satisfy f(·yz) = f(z) for all 7 E F and

cacompact since .7: has a cusp at ioo. Functions on I`\H can be considered as

opposite sides of 8.77. I` is co-finite since vol(I`\H) = ff dz : rr/3 < oo, but not

I" is generated by the translation z •-—> z+1 and the inversion z +—~> -1/z identifying

.7`={z€H||z|21,|x|§1/2}. (13)

the quotient l`\H, is the modular domain

main, which is together with appropriate boundary conditions a realisation of

is one of the simplest discrete subgroups of G. Its standard fundamental do

(12)1`=PSL(2,Z)= {(‘;;)€Z2"2 |ad—bc=1}/{if}

The modular group

Both the volume element and the Laplacian are invariant under the action of G.

CZ_+dZ ·· ZZ G. az . y € ()

az + b a b (Cd)acting on H by (fractional) linear transformations

G:PSL(2,IR): {Q;) ginm |ad-bc= 1}/{i1} (10)

The group of orientation preserving isometries is

that the first Fourier coefficient never vanishes. Otherwise one would deduce from OCR Output

for even cusp forms while in the odd case sines replace the cosines. It can be shown

11.:]

(20)fj(z) = > K;Rj(2rrny)cos(2rm:1:)

employment of periodicity in m. This leads to the Fourier expansion

Formal solutions of (18) can be obtained by a separation ansatz and the

with positive Rj which henceforth will be called eigenvalue as well.

with Aj —-+ oo as —> oo. The lower bound 311*2/2 [30] allows to write Aj = {5+ R?

(19);)

coefficients c,, are polynomials in the c,,’s with p $ n. OCR Output

Where c,,/,, has to be taken as zero if p is not a divisor of rz. It follows that the

Cup = Cncp — C,./p (26)

E.g., for n E IN, p prime, one has

d|(m,n)

Cmcn Z ) J Cmn/d2

The coefficients are not all independent but obey the Hecke relations

(24)z = °pf() —·———— pz . 1 §f(z+j)+f( ) (/5 ,:0 Pn = p prime

real functions. More explicitly, equation (23) with the definition (15) reads for

Since 'I`,, is self—adjoint Cn E IR for all n E IN causing normalized cusp forms to be

Tnf(Z)=c.1f(Z) \/ne1N. (23)

expansion (20), their eigenvalues turn out to be the Fourier coefficients,

Letting the Hecke operators act on normalized cusp forms given by the Fourier

where d(n) counts the number of divisors of n.

(22)c,,| f d(n)n1/4,

the Bessel function and the estimate [40]

Series (20) is absolutely convergent on H due to the exponential decrease of

with an error being smaller than any desired 6 > 0.

one can find a value 1'Emax = xm,x(R,e:) > R and take K,R(:v) = 0 for x > :1:,,,,.,,

dies off roughly exponentially as :1: ——+ oo. Consequently, for numerical purposes

turning point lies slightly below :1: = R. For x > R, K,R(x) remains positive and

of order (/21r/Rexp(—·1rR/2). The last extremum is positive and the following

:1: E (0,R) with a period approximately proportional to z and an amplitude

is real for real IB and R. For fixed R, K,R(:1;) oscillates about 0 in the interval

K;R(x) denotes the modified Bessel function of the third kind [1, 41] which

We remark that normalized cusp forms do not have an L2-norm of one.

C1 Z

is fixed to be

paper will be dealing with so—called normalized cusp forms whose first coefficient

relations like (25) that all other coefficients have to vanish, too. The rest of the

equivalent to (2 +j)/p. The point zi? is easily found by translations into the strip OCR Outputwhere 2; : pz and 2; 6 .7·` denotes the point in the modular domain which is l`

pf(/=°) = —f(Z'?) + f(Z") J v (31)1 "" < JF Qone is allowed to write

for any R > O as an equation for the unknown cp. Since f shall be l`—invariant

which hold for any z E 'H and any cusp form. Fixing z = 2 6 F we interpret (24)

We do this with the help of the eigenvalue equations of the Hecke operators (24)

The main question in this approach is how to determine E, for the actual R.

cusp form.

to vanish (numerically) as a function of z E 'H. The corresponding f will be a

candidates for eigenvalues. The correct eigenvalues are picked up by testing (I) ,( R)

Then by scanning the R—axis we are looking for zeroes R of ,0(R) which are

z(R) = f(-Z; EMR) — f(-1/Z; Ep, R) (30)

The basic idea is the following. Let zu E .7: be fixed and

Neumann) boundary conditions on the lower part |z| = 1 of GF.

An equivalent requirement would be the imposition of the correct (Dirichlet or

f(z) = f for all z E H. (29)1

Hence the only remaining condition for f to be a cusp form is

The Fourier expansion automatically assures translation invariance f (2) = f (2+1).

F is generated by the translation T(z) = z-I-l and the reflection $(2) = -1/z.

l`\'H, i.e. not a cusp form, since it is not necessarily l`—invariant.

function of the Laplacian on 'H. But generally it is not an eigenfunction of A on

resulting function f (z) = f(z;E,,,R) is square summable over .7: and an eigen

of ”fake” coefficients that could be inserted into the Fourier expansion. The

é’=(1,c2,c3,c.,,...)= (28)

vector

be any real vector whose components fulfil (22). Using (26) Ep determines the

CLP Z (C2, C3, C5, . .

Consider the Fourier expansion (20). Let R > O and

3 The algorithm

procedure started. We try to find new solutions by choosing randomly starting OCR Output

for the actual R"°`”, the next step is very important as well as for getting the whole

However, since it is impossible to know if the list of solutions of (32) is complete

cusp form is given by its Fourier expansion.

not. In the former case one has hunted up an eigenvalue, and the corresponding

R0 is found one has to check with different points z ¢ zo whether z(R0) = O or

detect by a modified secant method or a modified Regula Falsi. lf such a zero

holds. In this case one expects a zero of @20 in the interval [R°ld, R“"’] which we

@20 (Rold)®z0 (Rncw) S O

iteration. Each time the iteration converges one tests if

R = R“°“’ = R°ld + AR one takes each Qld as a starting value for a Newtongenerally has a couple of solutions Egld for a given R = R°ld. Going to the nextthe calculations looks like follows. System (32) is nonlinear and therefore one

ing values for the Newton iteration randomly. In more detail a typical step of

advance where to find its solutions it turns out to be effective to generate start

This system is solved with Newton’s method. Since one does not know in

contain numerical data.

j=0r . SKC(p,n) = $ L \/yi? K,R(2rrny§)cos(2rm:1:§)

SKC'(1, n) = \/§K;R(21mg]) cos(21m5:),

where the constants

p: 2,3,5,...,}), 1r(N) = 7r(P)

n=1

0 = c,,SKC(1, n) — SKC(p, , i (32)2; [

mial, i.e. nonlinear, system of equations for the prime coefficients

Truncating the Fourier series and exchanging summations yields the polyno

equivalent points.

and its translates just contain the points with largest imaginary part of all I`

2; = R, the convergence of (20) is better the larger y is. The modular domain

Since the Bessel function K;R($) is a monotonically decreasing function of xc above

number of terms needed to compute the Fourier series to a prescribed accuracy.

Rezl § 1/2 and reflections by z +——> -1/z in turn. This choice minimizes the

10 OCR Output

the linear algebra associated with Newton’s method. For this reason it was nec

Bessel function calls are the most time consuming part of the algorithm besides

arguments. The latter one is faster without loss of accuracy, but in any case the

formulae, as proposed in [8], combined with a Miller algorithm [38] for certain

of Hejhal and Bombieri [16], or our own routine which is based on asymptotic

For the evaluation of the Bessel functions we have been using either a routine

with R. At R z 500 we got typically about 50%.

and 100%. The portion of found eigenvalues in a single run is slightly decreasing

run found about 70 — 80% of the eigenvalues and all three together between 95%

simultaneously found solutions of (32) increases from about 20 to nearly 40. Each

11 prime coefficients at R = 100 and 19 at R = 250. The maximum number of

30 and each cp E [—2.2,2.2]. To yield an accuracy of 13 digits demands about

0.3 + i1.25. The R—grid was of width ii- (lg-), the number of random E, wasR E [100,250], the parity was odd. We took 2 = 0.2 + i1.1, 0.2 + i1.15, and

As an example we mention in more detail three of our runs in the interval

of eigenvalues.

values can increase the running time without an appropriate gain in the output

small values worsen the number of found eigenvalues in a single run and large

held between 10 and 30. Both quantities, F and N, have to be coordinated since

The number N of random E,,’s at each step of the computations was mostly

with F > 1. Good values of F turn out to be typically 10 < F < 30.

of f(z) — f(—1/z) should therefore have a (nonconstant) width of AR = dz}or odd eigenvalues of about 12/ R. The grid on which one searches for sign changes

From Weyl’s law (cf. (40,41)) one expects a mean distance between successive even

Some technical remarks

eigenvalues.

doing several runs with different values of 2 we were able to complete the list of

with distinct 2. For ”macroscopically” different 2 this seems to be true, and by

eigenvalues one can hope for the statistical independence of the solutions found

2 E T in eqs. (32) and (33). Since 2 carries no a priori information about the

To circumvent this problem there is still the freedom of choosing the parameter

With a single run one does usually not find all eigenvalues in a given interval.

(22) and Ramanujan—Petersson conjecture (5), respectively.

values for the iteration, i.e. typically cp E [-2pl/‘*,2p1/"] or cp Q [-2,2] due to

ll OCR Output

the problems at hand, but this is not true. The problem of convergence is simply

first sight the introduction of the nonlinear system (32) might seem to increase

Our method does not have these problems with stability and convergence. At

suited for detecting unknown solutions.)

still a kind of experimental stage. It deserves some supervision and is not well

given approximations. In spite of some tricky improvements their method has

Stark’s iterative method to compute the coefficients of certain cusp forms from

matrix. (To avoid this problem Hejhal & Arno [17] went back to the ideas of

involved. The reason of this behaviour is the ill-conditioning of the established

find the eigenvalues but not sufficient to get out correctly all the coefficients

The main drawback of this method is its lack of stability which is sufficient to

found by varying R and by the requirement that the Hecke relations be fulfilled.

which is solved for the unknown (not only prime) coefficients. The eigenvalues are

given by its Fourier expansion (20) and to deduce a system of linear equations

explicitly the equation f (z) — _f (-1 /z) = 0 for different points z E H where f is

[16]; it makes_use of the Hecke relations, too. The basic idea is to write down

The first method to find a larger number of eigenvalues stems from Hejhal

converging to this solution. Examples of this effect can be found in [17].

ing close to a correct eigenvalue and the corresponding E, the iteration is not

assure contraction of (35). It is easy to find examples in which, even if start

the first even eigenvalue, this method generally does not work since one cannot

where f]°(z) is given by the k-th iterate {c£],,€N. Although he was able to detect

(35)kcf.+*f = T. r*

coefficients by a direct iteration of equation (24) or slight rearrangements of it,

Maass wave forms is mentioned by Stark [36]. He proposed to compute Fourier

The idea to use Hecke operators in connection with computations concerning

4 Discussion of the algorithm

Most of the computations were carried out on IBM RS /6000 workstations.

relative to the argument is to be prefered.

for each argument 21rny? works well. For (fixed) large R a Newton interpolation

For small and moderate indices (R < 250, say) a Lagrange interpolation in R

essary to reduce drastically the Bessel function calls by interpolation techniques.

12 OCR Output

and 2.

running time is increasing enormously. The results are presented in the tables 1

R = 1000, 2000, and 4000. In either case eigenvalues were found although the

To explore the limits of the algorithm we have done a couple of test runs at

linear equations, whereas the presented method is not affected by this behaviour.

separation of such close lying solutions is difficult for any algorithm based on

[7, 18], and [10] displays somehow diplomatically the mean 23.23. Generally, the

eigenvalues 23.20+ and 23.26+ were not known correctly. They are missing in

extreme cases. It is visible even at small R: until recently [16] the two odd

3 clearly demonstrates the effect of nearly degenerate eigenvalues in the most

level clustering in the parlance of quantum chaos, as was shown in [4, 34]. Table

important in view of the fact that the eigenvalues tend to prefer small distances,

has not to be finer than the distance between successive eigenvalues. This is

existing. However, this possibility has not yet occurred. Secondly, the R—grid

makes it possible to detect without complications degenerated eigenvalues, if

eigenvalue in one mesh of the R—grid. This is good for two reasons. First it

The ambiguity of the non—linear system allows to find more than a single

where N z 825 and 1r(N) z 145 this leads to a factor of (825/145)3 z 184.

to the third power of the matrix dimension. In the extreme case of R z 4000

tiple solutions. Recall the expense of a Gauss elimination is roughly proportional

of (N -1) >< (N -1), in the linear case, compensates for the effort of computing mul

The reduced matrix dimensions of rr(N) >< rr(N), in the nonlinear case, instead

away from zero) the much smaller off-diagonal elements of the last columns.

term f (2) in each element of the diagonal which dominates (as long as it stays

ditioned, at least partly since differentiating (32) with respect to Ep produces a

tions. But in this case the relevant matrix of the derivative is much better con

The Newton method also reduces the nonlinear problem to solving linear equa

[-2 2*,2 -24] X [-2 -31/*,2-31/4] X [-2- 5I,2- 54] X ... (36)/41/ /41/

are contained in a finite box of dimension (see (22))

vergence. Its use is made easier by the fact that the solutions one is looking for

solved by application of Newton’s method which assures locally quadratic con

17 OCR Output

the mean very well from the very beginning, in spite of its asymtotic character.

such jumps we have completed our list. It should be noted that 1Vmean(A) describes

Accordingly, 6,, is shifted by 1 for all n 2 N. By searching eigenvalues around

an eigenvalue AN is missing, all subsequent n 2 N are underestimated by 1.

is the middle of the step at the n-th eigenvalue (if the spectrum is simple). lf

1000 eigenvalues of even parity. The bracket in (42) is equal to n -— 1/2 which

indicating the oscillatory behaviour about 0. As an example we took the first

(42)6¤ : Nm

18 OCR Output

far the most extensive (numerical) test.

All about 320000 prime coefficients we know agree with this conjecture, being so

pI S 2. (45)

Petersson conjecture asserts an improvement to

second one delivers smaller numerical bounds for moderate p. The Ramanujan—

of view the first estimate is the better one because ofthe smaller exponent but the

a fact we have been exploiting in the algorithm. From a number theoretical point

Cpl S 2P, (44)l/4

Icp! s 2 (pm + pom)

of the normalized cusp forms, obey the bounds [32, 40]

It is known that the eigenvalues of the Hecke operators, i.e. the Fourier coefficients

6 Eigenvalues of the Hecke operators

1000 even eigenvalues An as a function of n.Figure 1: Difference 6,, of the spectral staircase N(/\) to Weyl’s law for the first

0 100 200 300 400 500 600 700 800 900 1000

-2

l I° **l l ” lf1 , ¤ I .4 ii l l ll l 0 ll i 1 rl { lt _ l "

19 OCR Output

examples are listed in table 6. Figure 2 shows the numerical data together with

have done similar computations including more coefficients and larger R. The

been computed whose number is 1229. To improve the statistical significance we

With the exception of the last one, all corresponding c,,’s with p < 10000 have

and two even ones of larger eigenvalue (R = 47.9+, R = 125.31+) are considered.

direction were done in [17] where the respective first five even and odd cusp forms

it is interesting to get some numerical hints on its validity. First studies in this(c,f,)p€P are equidistibuted with respect to dp. Since this conjecture is unprovedwould be true which states that the prime coefficients of a single cusp formconsidering (c;’,)j€N’P€P. If this were the case the so-called Sato—Tate conjectureThe question arises if one is allowed to reverse the limits j —+ oo and p —> oo in

otherwise.(48)dmm) Z] -Sgt/4 — mzdx, [ac] < 2

As p —> oo, dp,,(x) converges to the so-called semicircle distribution

agreement.

given coefficients with the prediction for a couple of p’s. One recognizes a good

correctness of our results. Figure 3 compares the distribution of the numerically

upon p for small values of p. This delivers an additional overall test of the

The densities d;z,,(:c) are by no means trivial and show a strong dependence

lim = f(x)d/\(z) for all f 6 Cc[lR). (47)‘/ nqoo TL i=1 R

Equidistribution of a sequence (xj)jEN C IR with respect to d/\ means

0, otherwise.

P(+ 1)\/KFdx

with respect to

dered by increasing corresponding eigenvalues. Then they are equidistributedConsider the sequence MN, p prime, of coefficients with fixed index and or

that not too much coefficients can violate the Ramanujan—Petersson conjecture:

We turn now to statistical properties of the c,,’s. A result of Sarnak [32] shows

eigenfunction.

even R. This is equivalent to the numerical knowledge of the corresponding

Table 2 exhibits as a reference eigenvalues of the Hecke operators for a large

20 OCR Output

many helpful hints.

I would like to thank J. Bolte and F. Steiner for reading the manuscript and for

Acknowledgements

its argument and then to compute it by means of a simple interpolation.

time it is best first to tabulate the Bessel function in the whole needed range of

that reason n = N is only needed for the lowest lying points in F. To save cpu

distributed over the modular domain with respect to dz as p —> oo [36], and for

arguments for which 2vrnyf? < .1:,,,3,.. The points yl? are assumed to be randomly

Q is introduced due to the fact that one can restrict the evaluations to thosecp requires an effort of about QN p Bessel function evaluations. Here, a factor ofThe crucial point is the increasing computing time as R and p get large since each

p exceeds the truncation index N of the Fourier series (20), as proposed in [36].The method at hand is to take (24) as an explicit equation for cp whenever

Sato—Tate conjecture.

the semicircle law. The agreement is very good and a clear confirmation of the

Table 6: Eigenvalues for which the Sato—Tate conjecture is checked in figure 2.

aggregation of a. to e. 130000

5020222609522 odd 10000 104729

2499355630821 even" "

99.86259201648 even

13.77975].35189 even

9533695261354 odd 30000 350377

R Parity # cp z> S

21 OCR Output

ture. The small letters correspond to table 6.forms in comparison to the semicircle law as predicted by the Sato—Tate conjecFigure 2: Distribution of prime coefficients (Hecke eigenvalues) of certain cusp

l2 0 2 -2 0 2 `-2 0 2

0.1 0.10.1

0.2 0.20.2

0.3 0.30.3

0.4 0.40.4

Z2 0 2 l2 0 2 12 0 2

0.1 0.10.1

0.2 0.2 0.2

0.3 0.3 0.3

0.4 0.4 0.4

23 OCR Output

stam and C. Hooley, eds.), vol. 2, Academic Press, 1981, pp. 95-110.Dirichlet L~Series, in: Recent Progress in Analytic Number Theory HalberD. A. Hejhal, Some observations concerning eigenvalues of the Laplacian and[14]

Institut fiir Angewandte Mathematik, 1977.»\y`2u fdr ein unendliches Gebiet im IR2, Diploma thesis, Universitat Heidelberg,H. Haas, Numerische Berechnung der Eigenwerte der Dijferentialgleichung —Au =[13]

York, 1990.M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, New[12]

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