## Defining parameters

 Level: $$N$$ = $$816 = 2^{4} \cdot 3 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$26$$ Newform subspaces: $$84$$ Sturm bound: $$73728$$ Trace bound: $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(816))$$.

Total New Old
Modular forms 19328 7858 11470
Cusp forms 17537 7586 9951
Eisenstein series 1791 272 1519

## Trace form

 $$7586q - 22q^{3} - 48q^{4} + 4q^{5} - 16q^{6} - 32q^{7} + 24q^{8} + 2q^{9} + O(q^{10})$$ $$7586q - 22q^{3} - 48q^{4} + 4q^{5} - 16q^{6} - 32q^{7} + 24q^{8} + 2q^{9} - 48q^{10} + 24q^{11} - 32q^{12} - 60q^{13} - 24q^{14} - 4q^{15} - 96q^{16} - 2q^{17} - 80q^{18} - 16q^{19} - 32q^{20} - 48q^{21} - 96q^{22} - 16q^{23} - 88q^{24} - 34q^{25} - 40q^{26} - 46q^{27} - 64q^{28} + 20q^{29} - 72q^{30} - 80q^{31} - 48q^{33} - 56q^{34} - 48q^{35} - 64q^{36} - 12q^{37} + 16q^{38} - 60q^{39} - 16q^{40} + 12q^{41} + 56q^{42} - 48q^{43} + 80q^{44} - 12q^{45} + 32q^{46} + 96q^{48} - 54q^{49} + 72q^{50} - 58q^{51} - 64q^{52} + 4q^{53} + 64q^{54} - 16q^{55} + 72q^{57} - 96q^{58} + 40q^{59} - 48q^{60} - 60q^{61} + 24q^{62} + 104q^{63} - 192q^{64} + 152q^{65} - 136q^{66} - 32q^{68} - 8q^{69} - 208q^{70} + 112q^{71} - 104q^{72} + 52q^{73} - 104q^{74} + 158q^{75} - 192q^{76} + 96q^{77} - 128q^{78} + 80q^{79} - 16q^{80} - 22q^{81} - 112q^{82} + 168q^{83} - 16q^{84} - 60q^{85} + 32q^{86} + 84q^{87} + 12q^{89} + 16q^{90} + 80q^{91} + 32q^{92} - 8q^{93} + 16q^{94} + 112q^{95} + 80q^{96} - 76q^{97} + 80q^{98} + 104q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(816))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list the newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
816.2.a $$\chi_{816}(1, \cdot)$$ 816.2.a.a 1 1
816.2.a.b 1
816.2.a.c 1
816.2.a.d 1
816.2.a.e 1
816.2.a.f 1
816.2.a.g 1
816.2.a.h 1
816.2.a.i 1
816.2.a.j 1
816.2.a.k 2
816.2.a.l 2
816.2.a.m 2
816.2.c $$\chi_{816}(577, \cdot)$$ 816.2.c.a 2 1
816.2.c.b 2
816.2.c.c 2
816.2.c.d 2
816.2.c.e 4
816.2.c.f 6
816.2.e $$\chi_{816}(239, \cdot)$$ 816.2.e.a 4 1
816.2.e.b 8
816.2.e.c 20
816.2.f $$\chi_{816}(409, \cdot)$$ None 0 1
816.2.h $$\chi_{816}(407, \cdot)$$ None 0 1
816.2.j $$\chi_{816}(647, \cdot)$$ None 0 1
816.2.l $$\chi_{816}(169, \cdot)$$ None 0 1
816.2.o $$\chi_{816}(815, \cdot)$$ 816.2.o.a 4 1
816.2.o.b 4
816.2.o.c 4
816.2.o.d 8
816.2.o.e 16
816.2.r $$\chi_{816}(395, \cdot)$$ 816.2.r.a 4 2
816.2.r.b 4
816.2.r.c 272
816.2.s $$\chi_{816}(157, \cdot)$$ 816.2.s.a 2 2
816.2.s.b 2
816.2.s.c 68
816.2.s.d 72
816.2.u $$\chi_{816}(205, \cdot)$$ 816.2.u.a 64 2
816.2.u.b 64
816.2.w $$\chi_{816}(203, \cdot)$$ 816.2.w.a 280 2
816.2.y $$\chi_{816}(455, \cdot)$$ None 0 2
816.2.ba $$\chi_{816}(217, \cdot)$$ None 0 2
816.2.bd $$\chi_{816}(625, \cdot)$$ 816.2.bd.a 4 2
816.2.bd.b 4
816.2.bd.c 4
816.2.bd.d 4
816.2.bd.e 8
816.2.bd.f 12
816.2.bf $$\chi_{816}(47, \cdot)$$ 816.2.bf.a 8 2
816.2.bf.b 8
816.2.bf.c 8
816.2.bf.d 24
816.2.bf.e 24
816.2.bh $$\chi_{816}(35, \cdot)$$ 816.2.bh.a 4 2
816.2.bh.b 4
816.2.bh.c 124
816.2.bh.d 124
816.2.bj $$\chi_{816}(373, \cdot)$$ 816.2.bj.a 144 2
816.2.bl $$\chi_{816}(13, \cdot)$$ 816.2.bl.a 2 2
816.2.bl.b 2
816.2.bl.c 68
816.2.bl.d 72
816.2.bm $$\chi_{816}(251, \cdot)$$ 816.2.bm.a 4 2
816.2.bm.b 4
816.2.bm.c 272
816.2.bq $$\chi_{816}(49, \cdot)$$ 816.2.bq.a 8 4
816.2.bq.b 8
816.2.bq.c 8
816.2.bq.d 16
816.2.bq.e 16
816.2.bq.f 16
816.2.br $$\chi_{816}(287, \cdot)$$ 816.2.br.a 48 4
816.2.br.b 96
816.2.bs $$\chi_{816}(155, \cdot)$$ 816.2.bs.a 560 4
816.2.bt $$\chi_{816}(229, \cdot)$$ 816.2.bt.a 288 4
816.2.bw $$\chi_{816}(59, \cdot)$$ 816.2.bw.a 560 4
816.2.bx $$\chi_{816}(325, \cdot)$$ 816.2.bx.a 288 4
816.2.ca $$\chi_{816}(25, \cdot)$$ None 0 4
816.2.cb $$\chi_{816}(263, \cdot)$$ None 0 4
816.2.cf $$\chi_{816}(29, \cdot)$$ 816.2.cf.a 1120 8
816.2.cg $$\chi_{816}(91, \cdot)$$ 816.2.cg.a 576 8
816.2.cj $$\chi_{816}(65, \cdot)$$ 816.2.cj.a 24 8
816.2.cj.b 24
816.2.cj.c 32
816.2.cj.d 48
816.2.cj.e 72
816.2.cj.f 72
816.2.ck $$\chi_{816}(31, \cdot)$$ 816.2.ck.a 48 8
816.2.ck.b 96
816.2.cn $$\chi_{816}(7, \cdot)$$ None 0 8
816.2.co $$\chi_{816}(41, \cdot)$$ None 0 8
816.2.cr $$\chi_{816}(5, \cdot)$$ 816.2.cr.a 1120 8
816.2.cs $$\chi_{816}(139, \cdot)$$ 816.2.cs.a 576 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(816))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(816)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(68))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(136))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(204))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(272))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(408))$$$$^{\oplus 2}$$