We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of Lfunctions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.

We show, for levels of the form N = p a q b N ′ with N ′ squarefree, that in weights k ≥ 4 every cusp form f ∈ S k (N ) is a linear combination of products of certain Eisenstein series of lower weight. In weight k = 2 we show that the forms f which can be obtained in this way are precisely those in the subspace generated by eigenforms g with L(g, 1) = 0. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.1 When χ = 1N is the principal character modulo N we write M k (N ) for M k (N, 1N ).

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We prove an equidistribution result for the Satake parameters of the local representations attached to Siegel cusp forms of degree 2 of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of L-functions attached to these cusp forms. NotationThe algebraic group GSp 4 is defined as GSp 4 = {g ∈ GL 4 ; t gJg = λ(g)J for some λ(g) ∈ GL 1 }, where J = −1 2 1 2 .

We compute the Fourier coefficients of a basis of the space of degree two Siegel-Eisenstein series of square-free level N transforming with the trivial character. We then apply these formulae to present some explicit examples of higher representation numbers attached to non-unimodular quadratic forms.