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2-4. f We s h a l l prove Theorem 1. Set T(p). /2k. B Lemma 1, t h e r e e x i s t s a l i f t i n g Hecke operators. p So must be a cusp form. Let such t h a t 1 + x(p)p q sx f = RE* f of f k- 1 x (p) $ . Let a be an odd prime - ) which i s a common eigenfunction k-1 $ 0 + p k- 1 (mod L-1) , there (mod 11) : hence of a l l e x i s t s some F(E 2 s x) + F i r s t we assume 2). Then we have L 4 q. I t is clear t h a t f u n c t i o n with r e s p e c t t o q r 1 Put (mod 4) x(-1) = 1 = (-1) g = - f.

1, Then, f o r each such t h a t ai(p) k We assume (mod 0 E 2) (i) k a if i s an even, p o s i t i v e i n t e g e r . SE 0 (mod R) = q. 2 Let Suppose t h e r e e x i s t s a c e r t a i n - 1 + X(p)pk-l f o r every prime of t p. number s a t i s f y i n g k s 0 K be any prime d i v i s o r o f there exists a certain -2i(B k s x and be an odd prime f a c t o r of t h e numerator of p. Then 2 and (ii) k $ 0 Let R (mod R-1) be a prime d i v i s o r o f K be any prime if R f q, lying over R of f , (T) such t h a t I (mod ai(p) E 0 1) i s a prime f a c t o r of t h e numerator o f Bk,X' The proof of our theorems makes e s s e n t i a l use o f t h e following f a c t s : 1.

319)calculated by inspired by an earlier work of Iguscr [ ' j ] denotes the By Morita 'a genenil result,. %reonly at ti0,l,m the mcthod ofU01, dhich t = ( x ~ ) ~ / ( - ~ Jand c~), 24 auto~~~rphtsl~s of V and the w with inclice8 2,4,6 respceivdy. , p-aicwcles; -1. 8. 37, (1969). Nomalized period matrice8 I; Ann. Math. , Class rider of a definite quaternion with prime discriminant, o 44 (1958), 312-314. Hat. A ~ d Sci. Proc, Algebraic curves mod p and arithmetic groupr; Proc. aymp. pure math.

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