A352095 - OEIS

%I #15 Mar 11 2022 13:01:05

%S 0,0,0,0,0,0,1,1,3,4,6,9,14,17,27,34,46,61,82,99,135,165,208,261,325,

%T 389,490,584,708,852,1023,1200,1445,1687,1984,2327,2717,3133,3663,

%U 4199,4838,5557,6360,7225,8267,9344,10587,11968,13489,15126,17037,19023

%N Dimension of the space of Siegel cusp forms of genus 3 and weight 2n.

%C There are no nonzero Siegel cusp forms of genus 3 and odd weight.

%C Sequence satisfies linear recurrence of order 54 for a(n) when n > 57.

%H Andy Huchala, <a href="/A352095/b352095.txt">Table of n, a(n) for n = 0..20000</a>

%H O. Taibi, <a href="https://arxiv.org/abs/1406.4247">Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula</a>, arXiv 1406.4247 [math.NT] (2014), 64-65.

%H S. Tsuyumine, <a href="https://doi.org/10.2307/2374517">On Siegel modular forms of degree three</a>, Amer. J. Math., 108 (1986), 831-832.

%H <a href="/index/Rec#order_54">Index entries for linear recurrences with constant coefficients</a> signature (1,0,0,0,0,2,-1,-1,1,0,-1,-1,-1,2,-1,-2,2,1,0,0,-1,3,0,-3,2,0,0,0,-2,3,0,-3,1,0,0,-1,-2,2,1,-2,1,1,1,0,-1,1,1,-2,0,0,0,0,-1,1)

%F G.f.: p/q with p,q given in Sage program.

%F a(n) = A027634(n) - A029143(n).

%e The space of weight 18 Siegel cusp forms of genus 3 has dimension 4.

%o (Sage)

%o R.<x> = PowerSeriesRing(ZZ,100)

%o p = -x^56 + x^55 - x^54 - x^51 - 3*x^48 + x^47 - 3*x^46 - 2*x^45 - 2*x^44 - 3*x^43 - 4*x^42 - 2*x^41 - 7*x^40 - 3*x^39 - 8*x^38 - 4*x^37 - 10*x^36 - 6*x^35 - 10*x^34 - 9*x^33 - 9*x^32 - 9*x^31 - 13*x^30 - 5*x^29 - 15*x^28 - 6*x^27 - 11*x^26 - 10*x^25 - 8*x^24 - 8*x^23 - 11*x^22 - 4*x^21 - 10*x^20 - 5*x^19 - 6*x^18 - 5*x^17 - 6*x^16 - 2*x^15 - 6*x^14 - 2*x^13 - 3*x^12 - 3*x^11 - 2*x^10 - x^9 - 2*x^8 - x^6;

%o q = x^54 - x^53 - 2*x^48 + x^47 + x^46 - x^45 + x^43 + x^42 + x^41 - 2*x^40 + x^39 + 2*x^38 - 2*x^37 - x^36 + x^33 - 3*x^32 + 3*x^30 - 2*x^29 + 2*x^25 - 3*x^24 + 3*x^22 - x^21 + x^18 + 2*x^17 - 2*x^16 - x^15 + 2*x^14 - x^13 - x^12 - x^11 + x^9 - x^8 - x^7 + 2*x^6 + x - 1;

%o (p/q).list()[:30]

%Y Cf. A027634, A029143.

%K nonn

%O 0,9

%A _Andy Huchala_, Mar 09 2022