Some computational results on Hecke eigenvalues of modular forms on a unitary group

Generators for modules of vector-valued Picard modular forms

“…As the isotropy group of a cusp in S 4 is isomorphic to S 3 = {σ ∈ S 4 : σ(1) = 1} we find an action of S 3 on the Fourier-Jacobi expansion of a Picard modular form. This action is given by (X, Y, Z) → (−X, Z, Y ) for R 2 ∼ (34) (Finis' notation in [5], p. 153) and (X, Y, Z) → (X, ρY, ρ 2 Z) for R 3 ∼ (234).…”

“…§6. The Hecke rings Finis [5] analyzed the Hecke rings for the arithmetic groups Γ and Γ 1 [ √ −3]. These Hecke rings are the same outside 3 and are generated by operators T (ν), T (ν, ν) for ν ∈ O F with N(ν) = p, a prime congruent to 1 mod 3, and operators…”

“…Shintani [18] considered vector-valued Picard modular forms in an unpublished manuscript; in particular, he determined a criterion for such a modular form to be a Hecke eigenform in terms of the Fourier-Jacobi series. Explicit (scalar-valued) Picard modular forms were considered for F = Q( √ −1) by Resnikoff and Tai (see [12], [13]) and for F = Q( √ −3) by Shiga, Holzapfel, Feustel, and Finis (see [15], [6], [7], [4], [5], resp. ); in particular, for the latter case, Holzapfel [6] and Feustel [4] determined a presentation of the ring of scalar-valued Picard modular forms on the congruence subgroup…”

“…while Finis [5] computed a number of Hecke eigenvalues for low-weight (less than or equal to 12) scalar-valued forms. Vector-valued Picard modular forms have not attracted much attention so far.…”

“…where c runs over a complete set of representatives for √ −3O F /α √ −3O F . We refer to [5] and the literature given there.…”