Explicit Shimura's conjecture for \(\text{Sp}_3\) on a computer (Q2459307)

Let \(D_p(X)=\sum_{\delta=0}^\infty T(p^\delta)X^\delta\) be the generating series of Hecke operators for the symplectic group \(\text{Sp}_n\). Shimura conjectured that \(D_p(X)\) is expressed as \(D_p(X)=E(X)/F(X)\) with polynomials \(E(X)\) and \(F(X)\) in \(X\) with integral coefficients of degree \(2^n-2\) and \(2^n\), respectively. In the case \(n=2\), this was proved by himself. Shimura's conjecture for \(\text{Sp}_3\) was first computed by Andrianov. In this article, the authors give a different method to prove Shimura's conjecture for \(\text{Sp}_3\). Their result is based on the use of the Satake spherical map. The case \(n=4\) was treated by the second author [Math. Notes 81, No. 5, 605--608 (2007); translation from Mat. Zametki 81, No. 5, 676--680 (2007; Zbl 1201.11049)].