1.3. L-functions 11 Therefore, Θ4(q) = λf(q) + μg(q) for some constants λ, μ ∈ C. We may compare q-expansions to find the values of λ and μ: Θ4(q) = n≥0 S4(n)qn = 1 + 8q + 24q2 + 32q3 + 24q4 + · · · λf(q) + μg(q) = λ + μq + 24λq2 + 4μq3 + · · · . Therefore, it is clear that λ = 1 and μ = 8, so Θ4 = f + 8g. Since the expansions of f and g are easy to calculate (for example, using Sage see Appendix A.2), we can easily calculate the coeﬃcients of the q-expansion of Θ and, therefore, values of S4(n). The exact formulas given above for Sk(n), however, follow from some deeper facts. Here is a sketch of the ideas involved (the reader may skip these details for now and return here after reading Chapter 4): given Θ4 = ∑ cnqn and F (q) = ∑ ( ∑ d|n d)qn, one can find an eigenvector G(q) = ∑ bnqn for a collection of linear maps Tn (the so- called Hecke operators, Tn : M2(Γ1(4)) → M2(Γ1(4))) among spaces of modular forms, i.e., Tn(G) = λnG for n 1, and the eigenvalues λn = bn/b1 = ∑ d|n d. Moreover, the eigenvector G can be written explicitly as a combination of Θ4 and F . Finally, one can show that the coeﬃcients cn must be given by the formula cn = 8 ∑ d|n, 4 d d (see [Kob93], III, §5, for more details). 1.3. L-functions An L-function is a function L(s), usually given as an infinite series of the form L(s) = ∞ n=1 ann−s = ∞ n=1 an ns = a1 + a2 2s + a3 3s + · · · with some coeﬃcients an ∈ C. Typically, the function L(s) con- verges for all complex numbers s in some half-plane (i.e., those s with real part larger than some constant), and in many cases L(s) has an analytic or meromorphic continuation to the whole complex plane. Mathematicians are interested in L-functions because they are objects from analysis that, sometimes, capture very interesting algebraic information.

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