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MODULAR FORMS-page57 - 53 6.3 Let us give some examples...

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53 6.3 Let us give some examples. Example 6.1 . Let Δ( τ ) = η ( τ ) 24 . It is called the discriminant function. We know that Δ( τ ) satisfies (6.1) with k = 6 with respect to the group Γ = SL(2 , Z ). By (4.9) Δ( τ ) = 1 (2 π ) 8 ϑ 1 2 1 2 8 . Since Δ( τ ) = q Y m =1 (1 - q m ) 24 we see that the Fourier expansion of Δ( τ ) contains only positive powers of q . This shows that Δ( τ ) is a cusp form of weight 6. Example 6.2 . The function ϑ 00 ( τ ) has the Fourier expansion P q m 2 / 2 . It is convergent at q = 0. So ϑ 4 k 00 is a modular form of weight k . It is not a cusp form. Let us give more examples of modular forms. This time we use the groups other than SL(2 , Z ). For each N let us introduce the principal congruence subgroup of SL(2 , Z ) of level N Γ( N ) = { M = α β γ δ « SL(2 , Z ) : M I mod N } . Notice that the map SL(2 , Z ) SL(2 , Z /N Z ) , α β γ δ « ¯ α ¯ β ¯ γ ¯ δ « is a homomorphism of groups. Being the kernel of this homomorphism, Γ( N ) is a normal subgroup of Γ(1) = SL(2 , Z ). I think it is time to name the group Γ(1). It is called the full modular group . We have Lemma 6.2. The group
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