Nils, Fredrik and Nathan did the following:
As in Nils-Kohnen's paper we checked D_FF = const L(F,s,spin) for F a lift (using Fredrik's Poincare series code and Nils and Nathan's Siegel modular form code)
Then we computed D_FF for a nonlift (in particular, the unique wt 20 form that is not a lift). That wasn't so good:
- it's not a linear combination of L(F,s,spin) where F is a wt 20 eigenform
- the corresponding Dirichlet D_FF series doesn't have an Euler product
according to the Dirichlet series coeffs we generated for D_FF and David's code, we have a pole of residue of 7\cdot 10^14 at s=3/2
- know we're trying understand D_FF theoretically by comparing its construction to the construction of the Rankin-Selberg convolution in genus 1
With that floundering Nathan and Fredrik did the following
- Fredrik wrote his Poincare series code for Sage
Nathan started looking into computing the weight 35 cusp form \chi_{35} for Sp(4,\mathbb{Z}) via the paper of Aoki and Ibukiyama (theta constants) and Nathan's formula for \chi_{35} in terms of theta constants
Some things to try
- compute the inner product of upsilon20 with itself by approximating the integral