Some relations between the analytical modular forms and Maass waveforms for $PSL(2,\mathbb{Z})$

A. B. Venkov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 24, Tome 200 (1992), pp. 51-61

Résumé

The main goal of this paper is to prove that any even Maass cusp waveform $f$ up to a finite dimensional subspace is represented by some special series $$ f(z,\overline{z})=c+\sum_{k=2}^\infty a(k_1,k_2,k_3,m_1,m_2,m_3)y^k R^{k_1}(z)Q^{k_2}(z)S^{k_3}\overline{(z,\overline{z})R^{m_1}(z)Q^{m_2}(z)S^{m_3}(z,\overline{z})}\qquad{(1)} $$ where $6k_1+4k_2+2k_3=k=6m_1+4m_2+2m_3$ and $R(z)=E_6(z)$, $Q(z)=E_4(z)$, $S(z,\overline{z})+3/\pi y=E_2(z)$ are the analytical Eisenstein series, $c$, $a(k_1,k_2,k_3,m_1,m_2,m_3)$ are complex coefficients. The same representation (1) is true for any element $f\in\mathcal{H}$, $f(z)=f(-\overline{z})$, $z\in H$ the upper half plane, $\mathcal{H}=L_2(PSL(2,\mathbb{Z})\setminus H)$, up to a finite dimensional subspace, which may be ia trivial (see Theorem 2 and Remark in the end of the paper).