A165686 Dimension of the space of Siegel cusp forms of genus 2 and weight 2k which are not Saito-Kurokawa lifts of forms of genus 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 8, 11, 12, 14, 16, 19, 20, 24, 26, 29, 32, 37, 38, 44, 47, 51, 56, 62, 64, 72, 76, 82, 88, 96, 99, 109, 115, 122, 130, 140, 144, 157, 164, 173, 183, 195, 201, 216, 225, 236, 248, 263, 270, 288, 299, 312, 327, 344, 353, 374

Offset: 1

Author

Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009

Keywords

• nonn
• easy

Comments

Also the dimension of the largest Hecke-closed subspace of forms in S_k(Gamma_2) which satisfy the Ramanujan-Petersson conjecture. These forms are also characterized by the property that their (Andrianov) spinor zeta function does not have any pole.

Examples

			a(20)=1 because there is exactly one Siegel modular form of genus 2 and weight 20 which is not a lift of some form of genus 1.
		

References

• M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhaeusser, 1985.
• T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256(3), p. 323-340, 1981.
• R. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).

Crossrefs

Cf. A165684 for the full space of Siegel cusp forms. See also A029143, A027640, A165685.

Formula

For k > 1 we have a(k) = A165684(k) - A008615(2k-5).

Conjectured G.f.: -x^10*(x^7+x^6-x^2-x-1) / ((1-x^2)*(1-x^3)*(1-x^5)*(1-x^6)). - Colin Barker, Mar 30 2013