Unformatted text preview: ) is bounded on H (it is not true if f is not cuspidal). (iii) Show that the coeﬃcient a n in the Fourier expansion f ( τ ) = P a n q n can be computed as the integral a n = Z 1 f ( x + iy ) e-2 πin ( x + iy ) dx. (iv) Using (iii) prove that | a n | = O ( n k ) (Hecke’s Theorem). 7.12 Let L be an even unimodular lattice in R 8 k and r L ( m ) be deﬁned as in Example 7.2. Using the previous exercise show that r L ( m ) = 8 k B 2 k σ 4 k-1 ( m ) + O ( m 2 k ) . 7.13 Let L = E 8 ⊕ E 8 ⊕ E 8 . Show that θ L = 1 ζ (12) E 12 + 432000 691 Δ ....
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- Fall '09
- Algebra, Eisenstein, Modular form, 8k, Unimodular lattice, commutative graded algebra, explicit linear relation