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 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

%I #11 Mar 19 2024 10:25:12

%S 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,

%T 37,38,44,47,51,56,62,64,72,76,82,88,96,99,109,115,122,130,140,144,

%U 157,164,173,183,195,201,216,225,236,248,263,270,288,299,312,327,344,353,374

%N 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.

%C 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.

%D M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhaeusser, 1985.

%D T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256(3), p. 323-340, 1981.

%D R. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,0,0,1,-1,-2,-1,1,0,0,1,1,0,-1).

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

%F 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

%e 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.

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

%K nonn,easy

%O 1,12

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

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Last modified September 19 02:20 EDT 2024. Contains 376003 sequences. (Running on oeis4.)