A trace formula for hecke operators on L2(S2) and modular forms on Γ0(4)

Abstract

To any finite symmetric subsetR ⊂ O3 corresponds a Hecke operatorTR on L2(S2) which leaves the eigenspaces ℋn (n ≥ 0) of the Laplacian invariant. We compute the trace ofTR | ℋn and prove that the sum of the positive eigenvalues ofTR on ⊕k=0n-1ℋk prevails over the modulus of the sum of the negative eigenvalues. For anym ∈ ℕ the integral quaternions of normm define such a Hecke operator\(T_{R_m } \), and renormalizing the traces ofTRm | ℋn slightly, we obtain sequences of Fourier coefficients of modular forms on Γ0(4).