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MODULAR FORMS-page119 - 115 Corollary 11.3 Keep the...

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115 Corollary 11.3. Keep the notation from the previous lemma. Assume f is normalized so that c 1 = 1 . Then c m c n = c mn if ( m, n ) = 1 , c p c p n = c p n +1 + p 2 k - 1 c p n - 1 where p is prime and n 1 . Proof. The coefficient c n is equal to the eigenvalue of T ( n ) on M k (Γ(1)). Obviously c m c n is the eigenvalue of T ( n ) T ( m ) on the same space. Now we apply assertion (ii) taking into account that the correspondence R p acts as multiplication by p - 2 k and remember that we have introduced the factor n 2 k - 1 in the definition of the operator T ( n ). Example 11.2 . Let E k ( τ ) be the Eisenstein modular form of weight 2 k , k 2. We have seen in (6.21) that its Fourier coefficients are equal to c n = 2(2 π ) k σ 2 k - 1 ( n ) ( k - 1)! , n 1 , c 0 = 2 ζ ( k ) = 2 2 k - 1 π k B k (2 k )! . Thus c n = c 1 σ 2 k - 1 ( n ), and therefore E k ( τ ) is a simultaneous eigenvalue of all the Hecke operators. Corollary 11.4. σ 2 k - 1 ( m ) σ 2 k - 1 ( m ) = σ 2 k - 1 ( mn ) if ( m, n ) = 1 , σ 2 k - 1 ( p ) σ 2 k - 1 ( p n ) = σ 2 k - 1 ( p n +1 ) + p 2 k - 1 σ 2 k - 1 ( p n - 1 ) , where p is prime and n 1 . Example 11.3 . Let f = Δ. Since f spans the space of cusp forms of weight 6 and this space is T ( n )-invariant for all n , we obtain that f is a simultaneous eigenfunction for all the Hecke operators. We have Δ = q Y m =1 (1 -
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