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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.

%I A165686 #11 Mar 19 2024 10:25:12
%S A165686 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 A165686 37,38,44,47,51,56,62,64,72,76,82,88,96,99,109,115,122,130,140,144,
%U A165686 157,164,173,183,195,201,216,225,236,248,263,270,288,299,312,327,344,353,374
%N 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.
%C A165686 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 A165686 M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhaeusser, 1985.
%D A165686 T. Oda, On the poles of Andrianov L-functions, Math. Ann. 256(3), p. 323-340, 1981.
%D A165686 R. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula). Preprint, Mannheim (1993)
%H A165686 <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 A165686 For k > 1 we have a(k) = A165684(k) - A008615(2k-5).
%F A165686 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 A165686 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 A165686 Cf. A165684 for the full space of Siegel cusp forms. See also A029143, A027640, A165685.
%K A165686 nonn,easy
%O A165686 1,12
%A A165686 Kilian Kilger (kilian(AT)nihilnovi.de), Sep 26 2009