Modular forms and regular representations of groups of order 24 (Q1387321)

Theorem: Let \(G\) be a grup of order 24, let \(\Phi\) be its regular representation, and let \(P_g(X)= \prod_k(X^{a_k} -1)^{t_k}\) be the characteristic polynomial of \(\Phi(g)\) for \(g\in G\); then the eta product \(\eta_g(z)=\prod_k(\eta(a_kz))^{t_k}\) is an eigenfunction of all Hecke operators. It is known [\textit{D. Dummit}, \textit{H. Kisilevsky} and \textit{J. McKay}, Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)] that there are exactly 30 eta products \(\prod_j \eta(n_jz)\) with \(n_j\in\mathbb{N}\), \(\sum_jn_j=24\), which have multiplicative coefficients. Only 8 of them appear in the form \(\eta_g (z)\) for groups of order 24; a complete list is given. In an earlier note [Funct. Anal. Appl. 29, No. 2, 129-130 (1995); translation from Funkts. Anal. Prilozh. 29, No. 2, 71-73 (1995; Zbl 0847.11022)] the author treated the same problem for finite subgroups of \(SL(5,\mathbb{C})\).