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MODULAR FORMS-page118

# MODULAR FORMS-page118 - 114 LECTURE 11 HECKE OPERATORS From...

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114 LECTURE 11. HECKE OPERATORS From now on we shall identify M (Γ(1)) k with F k . So we have linear operators T ( n ) in each space M (Γ(1)) k which also leave the subspace M (Γ(1)) 0 k invariant. To avoid denominators in the formulas one redefines the action of operators T ( n ) on the vector space M (Γ(1)) k by setting T ( n ) f = n 2 k - 1 T ( n ) * ( f ) = n 2 k - 1 X A ∈A n f | k A (11.14) These operators are called the Hecke operators . Let T ( n )( X m =0 c m q m ) = X m =0 b m q m . (11.15) It follows from (11.9) that for prime n = p , we have b m = ( c pm if p | m , c mp + p 2 k - 1 c m/p if p | m . (11.16) Also, for any n , b 0 = σ 2 k - 1 ( n ) c 0 , b 1 = c n . (11.17) 11.4 We will be interested in common eigenfunctions of operators T ( n ), that is, functions f ∈ M k (Γ(1) satisfying T ( n ) f = λ ( n ) f for all n. Lemma 11.3. Suppose f is a non-zero modular form of weight 2 k with respect to Γ(1) which is a simultaneous eigenfunction for all the Hecke operators and let P c n q n be its Fourier expansion. Then
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