We give some explicit eigen-values of Hecke operators for the cusp forms of weight 2 for the principal congruence subgroups Г(q) for several primes q, using the Eichler\u27s formula [1]. For the prime q≡3 mod 4, the cusp forms correponding to the degree q-1 representation of the group SL_2 (Z/qZ) are viewed as the cusp forms for Г_0 (q^2). Using these cusp forms, we give an example of the class field over the imaginary quadratic field Q (√) for q=43 which is constructed by Shimura\u27s theory of class fields over the quadratic fields

Yamauchi Masatoshi

山内 正敏

Kansai University Repository

Some Hecke Eigen-Values for Г(q)著者 Yamauchi Masatoshijournal orpublication title情報研究 : 関西大学総合情報学部紀要volume 26page range 73-86year 2007-01-10URL http://hdl.handle.net/10112/11876関西大学総合情報学部紀要「情報研究」第26号Some Hecke Eigen-Values for r⑩ Masatoshi YAMAUCHI* Abstract We give some explicit eigen-values of Hecke operators for the cusp forms of weight 2 for the principal congruence subgroups r(q) for several primes q, using the Eichler's formula [l]. For the prime q三 3mod 4, the cusp forms correponding to the degree q-1 representation of the group SL2(z / qz) are viewed as the cusp forms for r。(q2).Using these cusp forms, we give an example of the class field over the imaginary quadratic field Q (汽） for q = 43 which is constructed by Shimura's theory of class fields over the quadratic fields. * Faculty of Informatics, Kansai University 74 関西大学総合情報学部紀要「情報研究」第26号 2007年1月Introduction In this paper, we give some explicit eigen-values of Hecke operators for the cusp forms of weight 2 for the principal congruence subgroups r(q) for q = 23,31,43 and 47 corresponding to the representation of degree q-1 of the group SL2 (Z /q Z),using the Eichler's formula [l]. For the prime q三 3mod 4, the cusp forms correponding to degree q -1 representations are viewed as the cusp forms for r。(qり， andcontained in the space 81 or Sm defined in [3]. The cusp forms corresponding to the representations of degree q or q+ 1 are viewed as forms of r。(q)with some character of conductor q, of which Hecke eigen-values are not treated here, since they are rather easily obtained. We denote by Z , Q , 罠 andC the ring of rational integers, the rational number field, the real number field and the complex number field, respectively. Gら（股） denotes the group of all non-singular square matrices whose components are contained in 股 andSL2 (Z) denotes the group of all square matrices of determinant 1 whose components are contained in Z . 1 Cusp forms for r(q) and r0(砂）Let a = (: ~) E G L鵡） (det(a) > 0). For a function f(z) on the complex upper half plane SJ and a positive integer k 2 2 , we define a function JI [a]k on SJ by Ul[a収）(z) = (detal12(cz + d)-k f(a(z)), where a(z) = (az + b)/(cz + d). For a positive integer N, we define ro(N) = { (: !) E Sら(Z)I c = 0 mod N} . r(N) = { (: ~) E SL心） I (: ~)三(~ 『)modN}. We put r = r(l)= SL立） • We denote by Skげo(N))(resp.S直 (N)))the space of all holomorphic cusp forms of weight k on SJ satisfying fl[叡=f for r Er。(N) (respS(N)) Further for a characterゆ moduloN, we denote by S亭 o(N),ゅ） the space of all holomorphic cusp forms of weight k satisfying a b fl[叡＝心(d)f for r = (c d) E丘(N).If心isthe identity character, we simply write Skげo(N),心） as skげo(N)) . Hereafter we take N = q,q is an odd prime. We put研=r /r(q) . Throughout this paper we assume k = 2.Let 6 = S2げ(q) be the space of cusp forms of weight 2 for r(q). Some Hecke Eigen-Values for I'(q) 75 Then it is known by Hecke [2], the space 6 is decomposed as (1) 6 = I: sq-l,μe,qーI,μ+L sq+I,ve,q+I,v十s心＋・・・?? ッwhere 6qーl,μ,etc. are irreducible components of 6 of the representation S of叩defined by （口~J oM~S(M) (口(~J-Here {1P1(z)・ ・ ・1Pm(z)} is a basis of 6 and 叫z)o M = (叫[M]2)(z).The positive integer sqーl,μ,is the multiplicity of e;q-l,μ, . The right upper index q-1 of 6qーl,μ, for example, means the degree of the representation andμwill be explaind later and exact values of sq-l,μ, etc. is given in 3. Tables (Table (A)). We have by (1), (2) a(M) = L sq-l,μ, びqー1,μ,(M)+ L 8q+l,v aq+l,v(l¥lI) + .. , ??ツwhereび，びq-l,μ,etc. denote the traces on the spaces 6, eq-I,μ, etc. Suppose n is a quadratic residue mod q, and take t so that n = t2mod q. For the Hecke operator T(n) acting on 6, define T(n) as 叫 =T(叫 whereUt E r and Ut 三 (~tり1) mod q. We put'I= { T(n) I () } x n = 1 , where X denotes the real primitive character modulo q of order 2. Since 加）M=M加）， forME r, the traceぴ（加）） of the representation of'I becomes (3) ぴ（加）） = (qー i)I:びqーI,μば(n))+ (q + 1) I: 砂q+I,vば(n))+ .・, whereびq-1,μ, q+l v び',etc. denote the traces on certain subspaces ilqーI,μ,ilq+l,v, etc. of 6 , and dimilqーI,μ=sqーI,μ'dimuq+l,v=sq+l,v'etc (Eichler[l]). In this paper, we exclusively treat the representation of苅 ofdegree q -1. This representation is characterized by the character x(S) =一((μ+(;: りwhereS ia an element畑 oforder q;1 and弘 isa q; 1 -th root of unity which is not eq叫 to土 1and 1 :s;μ:s; 号 inthe case q = 1,5 mod 12 and 1 :s; μ:s; デ inthe case q三 7,11mod 12 (Hecke[2]). Now take 76 関西大学総合情報学部紀要「情報研究」第26号 2007年1月00 t(z) = I: 心e21r~nz E S虹 (q))'n=l which is a common eigen-function of Hecke operators T(n). Then we know by Shimura([4], prop.3.53, prop.3.64) CX) f(qz) = I: 心e27rinzE S直 o(q2),r.p) n=l for a suitable character (fJ modulo q, which is a common eigen-function of Hecke operators. Further, for another character~modulo q, 00 !c;(q;) =~ 如）心e21rinzE S直 o(qり記），n=l fe (qz) is also a common eigen function of al Hecke operators. Further assume q is a prime such that q三 3mod 4. Hence if we put~= <pデ， then忍 =l.Thus any function f(z) E S2(I'(q)) may be regarded as a function of f(z) =た(qz)E S遣 o(qり）．Using the trace formula of Eichler [l], we give several characteristic polynomials q>J,n(x) of Hecke operators T(n) on the subspace of S2(I沿(q2))corresponding to the primitive functions }(z) for q = 23,31,43,47. We denote by X the non-trivial quadratic character of conducter q. Then for J(z) =~~=1a砂21rinz E 32げ(qり）， fx(z)= ~~=1x(n)ane如inz is also contained in S2 (r。(qり）．Let・S乳r。ぽ）） denote the subspace of al new forms in S紅 o(qり） • We put ふ(I'o(q))X= {fxlf E S2(I'o(q))} , then the space S2(I'o(q))文 isthe subspace of S似I'o(qり）．We denote by s; げo(qり） the orthogonal complement of S紅 o(q))Xin sgげo(qり） with respect to the Petersson inner product. Then we know by [3], s; げo(qり） decomposes into four spaces, (4) Sf (恥(qり） = 81④ Sn④ Snx ffism where S1 = {f E S2(I'。(q2))I JIW = f, fl公W=fl公｝Sn= {f E S2(I訊q2))I JIW = f, f応W=-fl公｝Bux = {f E S2(I'o(q2)) I JIW = -f, f応W=fl公｝Sm= {f E S2(I'o(q2)) I JIW = -f, fl公W=-fl公｝．Here W= (> ~1), and for the quadratic character X , o isthe twisting q X operator 1 q-1 応＝可I:x(u)Jl[a叫 for 知=(6~) u=l where g(x) is the Gaussian sum for X. Some Hecke Eigen-Values for I'(q) 77 We list the dimensions of the space Si, Sn and Sm in the table (B). (we note dim Sn= dim汎）．2 The case q = 43. In [3, §5], we have given an example of the form in S2げo(qり） which relates to the theory of Shimura's construction of class fields over real quadratic fields for the case q2 = 192. In this section we add one example for the case q2 = 432. Hereafter we put q = 43. We denote by OL the maximal orders of the number field L. Let f(z) = 立記1rinzbe a primitive form corresponding to the first factor of the Table (C) (3.a), n=l then we see a2 = ,/因，au=<-5, 0, 5, 0, -1>, 釘3=<2,-5, -5, 2, l>, a17=< -2, -8, 3, 2, -l> whereあ=<2,-3, 0, 1, O>. We note that the absolute norm ofあ is1. Here < u0 , u1 , u2 , u3 , u4 > denotes u0 + u戸 22+u四届+U3CT羞＋四心， whereび22=伍＋冨 witha primitive 22-th root of unity (2 . Let K be the subfield of <C generated by an for all n. 00 Now f() and its companions name ly『(z)=区咋e21rinz for all isomorphismびn=l from K into <C span IO-dimensional subspace of S亨 o(43り） • From this f(z) and its companions, we obtain an abelian variety of dimension [K: (Q ] (= 10) defined over (Q and an isomorphism 0 of K into End(A)c(Q, 0(a) is rational for all a E K. Further, we see that there is an automorphism P of K, other than the identity map, such that x(n)an = a~for all n. Thus, let F be the invariant subfield of K under p , then we have [K: Fj=2. Let c = (あ）o F be the ideal in F, and b =冨oK be the ideal in K. Then b2 = c and both the absolute norm of b and c is 1. Further we seeあ=<2,-3, 0, 1, O> = <2, 1, 0, 0, O> <l, -2, 1, 0, O> and <l, -2, 1, 0, O> is a unit. So we may put c =(2+ 唸）．We know by [6, prop.8 and prop.9] there exists an endomorphism rJ of A which is defined over the quadratic field k = (Q(v可） and satisfies the following conditions: (i) が＝一rJif c is the generator of Gal(k/(Q). (i) rJ2 = X(-l)q・idA. (ii) rJo0(a)=0(aP) OrJ for every a EK. For the ideal b given as above, we put ~= {t E Al 0(b)t = O} , then~is isomorphic to (o K /年 asan o k -module. further we put り={t E~I (rJ-0(1))t = O} , 3={tE~I (rJ+0(1))t=O}. 78 関西大学総合情報学部紀要「情報研究」第26号 2007年1月Now as in [5,Prop.9.2], we see The submodulesりandJ are o F-isomorphic to o F / c and t =り〶 J . Let k(t) (resp. k(tJ)) denote the smallest extension of k over which the points of F (resp. り） are rational. Then k(t) ia an abelian extension of k, and making Gal(k(t) / k)act onりandJ, we obtain an injective homomorphism Gal(k(t)/k)→ (op/c)x x (op/c)x. We shall examine the abelian extension k (り）/ k following the method explained in [3, §5]. Proposition 2.1 The field k(tJ) is a ray class field over k of conductor (yだ二厄 witha prime factor (of 1, one has r((a)) =外a)μ(amod() for every a in k prime to 43 [, whereμis the isomomorphism of Ok八ontoop/c and'P is a homomorphism of (o豆(v二厄））x into (op/cげ oforder 2. proof. Since every prime factor of the conductor f of k(り）/k divides 43・N([) = 43・11, [4, §7.5 and Prop. 7.23] we may put f = I;pfp where p runs all prime factors of 43・11 By the same argument as in the proof of [5, Th.2.3], we obtain r((m)) = x(m)μ(m mod [) for every m E z prime to 43. 11. Thus [k (tJ): k] = 5 or10, and so the [ exponent Jc = 1 by [4,Lemma 3.2]. To determine fe, we use the following fact which is nothing but [5, Th. 2.8] for the present c~se. Let l be a prime which divides N(() but not q = 43. Suppose x(l) = 1, and az is prime to C = (2+び22).Then f isdivisible only one prime factor of l.Put l = l・reand suppose [ divides f . Then r化） = azmod c. Take l = 1. Then Since an = < -5, 0, 5, 0, -l> is prime to c = (2十び22),we have fie = 0. So we may put戸爪／二冦 . By the Hasse's conductor ramification theorem, we find t = I and t'= I, since [ k(り）： k] = 5 or10. Thus we have r((a)) = 9っ(a)μ(amod() for every a in k prime to 43 (, with a homomorphism r.p of (o豆(vコ句）x into (op/cY and the isomorphismμof ok八ontoop/c. Now the orders of (oり(v二話）） Xand (op/cY are 42 and 10 respectively, and孤m)=・x(m) for rational m prime to 43. 11, so the order of r.p is 2. Further, r化） = an三ー1mod (2 + cr22) , we have [ k(り）： k]=lO. This completes our proof. Remark. If p is a rational prime such that x(p) = 1 then p = a 'for an integer a Some Hecke Eigen-Values for r(q) 79 E Q(汀） • We take a so that x(a +a')= -1. Then we have ap = a + a'mod (2十び22).3 Tables Table (A) In the Table(A), the mulplicity sq-I,μ, of eq-I,μ, etc. is given in the decomposition of 6 . (see the equality (1) in the text) 1. the table of sq-l,μ μ 三 Omod6 μ 三 3mod6 μ 三 1,5mod 6 μ 三 2,4mod 6 q = 1 mod 12 (q -1)/12 (q -1)/12 (q -1)/12 (q -1)/12 q三 5mod 12 (q + 7)/12 (q + 7)/12 (q -5)/12 (q -5)/12 q三 7mod 12 (q + 5)/12 (q -7)/12 (q -7)/12 (q + 5)/12 q = 11 mod 12 (q + 13)/12 (q + 1)/12 (q -11)/12 (q + 1)/12 q-l q-3 where 1 :; μ::; 4 in the case q = 1, 5modl2 and 1 :; μ::; 4 in the case q 三 7,11 mod 12. 2. the table of 3q+l,v u三 Omod6 u三 3mod6 v = 1,5 mod 6 u三 2,4mod 6 q三 1mod 12 (q -25)/12 (q -13)/12 (q -1)/12 (q -13)/12 q三 5mod 12 (q -17)/12 (q -5)/12 (q -5)/12 (q -17)/12 q = 7 mod 12 (q -19)/12 (q -19)/12 (q -7)/12 (q -7)/12 q = 11 mod 12 (q -11)/12 (q -11)/12 (q -11)/12 (q -11)/12 q-5 q-3 where 1~v さて「 in the case q三 1,5 mod 12 and 1~v~ 4 in the case q = 7, 11 mod 12. Remark that 3q+l,v = dimS紅 o(q),心）， whereゆiaa Dirichet character of conductor q and of order t. t is determined as (v is a primitive t-th root of unity,where (is a . . q-l pr1m1tive 2 -th root of unity. 3. the table s号 ? ? ?＝??ー ???(q-25)/24 q三 1mod 24 (q -5)/24 q三 5mod 24 (q-17)/24 q三 17mod 24 (q -13)/24 q = 13 mod 24 80 関西大学総合情報学部紀要「情報研究」第26号 2007年1月Remark that in this case, q三 1mod 4, and sデ=!dimS叫 (q,(-), where (-) denotes 2 q q the Legendre symbol of level q. 4. the table s号Sデ =½h(-q)+ { (q + 5)/24 q三 7mod 24 (q + 1)/24 q三 11mod 24 (q -7)/24 q三 19mod 24 (q + 13)/24 q三 23mod 24 In this case, q三 3mod 4, and h(-q) denotes the class number of the imaginary quadratic field Q(汽）．5. the table sq s q ＝ ｛ (q-25)/24 q三 1mod 24 (q -5)/24 q三=5 mod 24 (q + 7)/24 q三=17 mod 24 (q -13)/24 q三 13mod 24 Note that呼=dimS2(ro(q)). Table (B) In this Table(B), dimSぃdimSn(= dimSux) and dimSm are given in the decomposition of S~(r0(q2)). (see the equality (4) in the text) dimS1 dimSn dirnSm q三 1mod 12 (q2 -22q + 117)/48 (q2 -2q + 1)/48 (q2 -6q + 5)/48 q三 5mod 12 (q2 -22q + 85)/48 (q2 -2q -15)/48 (q2 -6q + 5)/48 q三 7mod 12 (q2 -6q -7)/48 (q2 -l4q + 49)/48 (q2 + 2q -15)/48 q = 11 mod 12 ぽ— 6q-7)/48 （砂— 14q+ 33)/48 （砂+2q-47)/48 Some Hecke Eigen-Values for r(q) 81 Table (C) Table (C) is the table of the product 4》J,n(x)・4り;,n(x).where f(z) belongs to the space uq-l,μ. (La) q=23, μ= I orμ= 5 (contained in S1) T(2) (x + 1)2. (x + 1)2 T(3) (x -(-1 + a))2 T(5) (x -(a/3 -f3)/2)(x + (a/3ー /3)/2) T(7) (x + af3)(x -a/3) T(ll) (x -(a/3 + 2f3))(x + a/3 + 2/3) T(l3) (x + l + 2a)2 where a = v'3, /3 = v'2 . (Lb) q = 23,μ= 2 (contained in Sm) T(2) (x -a:)2. (x + a:)2 T(3) (x -(1 + a:))2・(x -(1 -a:)2 T(5) (x -a)(x +a)・(x -a)(x + a) T(7) (x -(-3 + a:))(x + (-3 +a:))・(x -(3 + a:))(x + 3 + a:) where a: = V3. (Le) q = 23, μ= 4(contained in Sm) T(2) (x -(1 + /3))2. (x -(1 -{3)2 T(3) (x -{3)2. (x + {3)2 T(5) (x -(1 -2/3))(x + (1 -2/3))・(x -(1 + 2f3))(x + 1 + 2/3)T(7) (x -(2 + f3))(x + 2 + /3)・(x -(2 -f3))(x + (2 -/3) where f3 =辺ふ82 関西大学総合情報学部紀要「情報研究」第26号(1.d) q=23, μ= 3 (contained in S1) T(2) T(3) T(5) T(7) (x -(-1 + ,y)/2)2・(x + (1 + ,y)/2)2 (x + 1)2・(x + 1)2 (x -(a,y -a)/2)(x + (a,y -a)/2)・(x -(cじ'Y+ a)/2)(x + (a,y + a)/2) (x + (a,y + a)/2)(x -(cじァ十a)/2))・(x+ (a,y -a)/2)(x -(a,y -a)/2)) where a = V:ふ'"'(=VJ図(2.a) q=31, μ= 1 orμ= 3,5, 7 (contained in S1) T(2) j (x2 -a四＋妬）2T(3) I x4 -a3丑+b3 T(5) I (x2 -a5x十転）2where a2 = < -1, -1, 0, O>, 妬＝＜ー1,1, 0, O> a3 = <6, -1, 0, l>, b3 = <4, 4, 1, O>, a5 = <O, 3, -1, -1>, bs = < -5, 1, 2, O> and <u0, u1, u2, u3 > denotes u0 + u位 16+U四む+u虹心， whereび16= (16 + (計 witha primitive 16-th root of unity (16. (2.b) q = 31, μ= 2 orμ= 6 (contained in Sm) >• 1:>~(~~~::::~:~:~~~ニ： \;~; ニ(3)where (3 = h. (2.c) q=31, μ=4 (contained in Sm) T(2) (x + 1)2・(x2 -3x + 1)2 T(3) (x2 -8)・(x4 -6叶+4)T(5) x2 . (x2 -5)2 2007年1月Some Hecke Eigen-Values for r(q) (3.a) q = 43, μ= 1 orμ= 5,7,9 (contained in 81) T(2) (x2 -四） ・(x4-b2庄＋叫T(ll) (x -au)2・(x2 -bux + cu)2 T(l3) (x -a13)2・(x2 -b13x + C13)2 T(l7) (x -a17)2・(x2 -b11x + c11)2 where a2 = <2, -3, 0, 1, O>, 的=<4, -1, 6, 1, -2>, C2 = < -1, 7, 17, -1, -5>, au = < -5, 0, 5, 0, -1>, b1 = <8, 4, -19, -2, 5>, C1 = <9, 18, 3, -6, -2>, a13 = <2, -5, -5, 2, l>, b13=< -5, 5, 9, -2, -2>, C13 = < -12, -6, 18, 2, -5>, a17 = <-2, -8, 3, 2, -1>, b17 = <l, -2, -4, 1, l>, 83 c17 = <-31, -15, 39, 7, -10> and <u0, u1, u2, u3, u4> denotes u0+u戸22+u匹品＋U30'羞+u4び五， whereび2=伍 ＋ ＜立 witha primitive 22-th root of unity (22. (3.b) q = 43, μ=2 orμ= 4,6,8,10. (contained in Sm) T(2) x8 -a2砂＋妬X4-C2丑+d2T(ll) (x4 -an討+bnx2 -C11X + d11)2 T(l3) (x4 -a13庄+b13X2 -C13X + d13)2 where四=<17, 0, -8, 0, 2>, 的=<90, -3, -72, 1, 18>, C2 = <154, 2, -149, -1, 36>, 必=<40, 17, -25,・-6, 4>, an = <9, 2, -8, 0, 2>, b口=<13, 14, -33, -2, 8>, en = < -80, -32, 147, 1, -40>, dn = < -94, 36, 72, -12, -14>. a1s = <3, 4, 0, 0, O>, bis = < -15, 9, 13, 0, -3>, C13 = < -10, -10, -36, -11, 14>, d13 = <9, -50, 13, 13, -14> and <u0, u1, u2, u3, u4 ,> denotes u0 + u10'11 + u匹五+u3心+u紅 f1, where 0'1 = (1 + 伍1with a primitive 11-th root of unity (11. 84 関西大学総合情報学部紀要「情報研究」第26号(4.a) q = 47, μ= l orμ= 5,7,11 (contained in S1) T(2) T(3) T(5) (x3 -a2丑＋妬X- C2)2 (x3 -a3砂＋妬X- C3)2 砂 -as企+b5炉— C5where四＝＜ー1,-1, 0, O>, 妬＝＜ー2,1, 0, O>, C2 = <O, 1, 0, O>, a3 = < -1, -4, 0, l>, 妬＝＜ー5,4, 1, -1>, C3 = <O, 15, 0, -4>, as = <14, 3, 0, O>, 転=<40, 8, 2, 4>, C5 = <4, -12, 14, 11>. 2007年1月< u0 , u1 , u2 , u3 > denotes u0 + u位 24+u心 +u3羹 where び24=如＋冨 witha primitive 24-th root of unity (24 . (4.b) q = 47, μ= 2 orμ= 10 (contained in Sm) □ I~::+:(口：口；:.+:~:: ロ::;x~4:(三！ニー(:1~:)~;: (613 -1400) where a = v'3. (4.c) q = 47, μ= 4 (contained in Sm) T(2) (x4 -3x3 -丑+6x -1)2 T(3) (x4 -2x3 -5丑+7x -2)2 T(5) x8 -30x6 + 279x4 -837x2 + 81 (4.d) q = 47, μ= 8 (contained in Sm) T(2) T(3) T(5) (x + 1)2・(x3 -2x2 -3x + 5)2 (x -2)2・(研+2x2 -5x -5)2 (x + 3)(x -3)・(x3 + x2 -4x + l)(x3 -x2 -4x -1) (4.e) q = 47, μ= 3 orμ= 9 (contained in 81) T(2) (x4 + (1+ f3)x3 + (-5 + f3)x2 + (-3 -4/3)x + (3-3/3))2 T(3) (x4 + (2-f3)x3 + (-5 -2/3)x2 + (-6 + 4/3)x + (3+ 5/3))2 T(5) x8 + (-26 + 9/3)x6 + (260 -152/3)x4 + (-944 + 656/3)x2 + (612 -432/3) Some Hecke Eigen-Values for I'(q) where (3 = V2. (4.f) q = 47, μ= 6 (contained in Sm) T(2) T(3) T(5) Table (D) (岱 -8企+8ぉ+3)2 (x5 -2x4 -7企+12笠+7x -8)2 x10 -36企+460x6 -2416企+4432x2 -2592 Table (D) is the table of q>f,n(x) where f(z) belongs to the space l芳．l.q = 23, h(-23) = 3 (contained in Sm) ロ2. q = 31, h(-31) = 3 (contained in Sm) T(2) x3 -6x -1 T(3) 砂T(5) x3 -l5x -2 3. q = 43, h(-43) = 1 (contained in S1) T(2) T(ll) T(13) X・(x2 -6) (x + 1・(x+ 1)2 (x-3)・(x+3)2 4. q = 47, h(-47) = 5 (contained in Sm) T(2) x5 -10砂+20x-9 T(3) x5 -15企+45x-28 T(5) 砂85 86 関西大学総合情報学部紀要「情報研究」第26号 2007年1月References [ 1 ] Eichler, H. Einige Anwendungen der Spurformel im Berich der Modularkorrespondenzen, Math.Ann., 168 (1967), 128-137. [ 2 ] Hecke, E. Grundlagen einer Theorie der Integralgruppen und der Integralperioden bei den Normalteilern der Modulgruppe, Math.Ann., 116 (1939), 469-510. [ 3 ] Saito, H. and Yamauchi, M. Trace formula of certain Hecke operators for r o(り）， NagoyaMath.J., 76 (1979), 1-33. [ 4] Shimura, G. Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten and Princeton Univ.Press, 1971. [ 5] Shimura, G. Class fields over real quadratic fields and Hecke operators, Ann.of Math., 95 (1972), 130-190. [ 6] Shimura, G. On the factors of the jacobian variety of a modular function fields, J.Math. Soc.Japan, 25 (1973), 523-543. 

Some Hecke Eigen-Values for Г(q)

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Some Hecke Eigen-Values for Г(q)

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