## Defining parameters

 Level: $$N$$ = $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$53$$ Sturm bound: $$26880$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(429))$$.

Total New Old
Modular forms 7200 5095 2105
Cusp forms 6241 4703 1538
Eisenstein series 959 392 567

## Trace form

 $$4703q + 9q^{2} - 35q^{3} - 55q^{4} + 18q^{5} - 39q^{6} - 80q^{7} - 31q^{8} - 59q^{9} + O(q^{10})$$ $$4703q + 9q^{2} - 35q^{3} - 55q^{4} + 18q^{5} - 39q^{6} - 80q^{7} - 31q^{8} - 59q^{9} - 142q^{10} - 9q^{11} - 159q^{12} - 117q^{13} - 36q^{14} - 84q^{15} - 183q^{16} - 42q^{17} - 107q^{18} - 156q^{19} - 106q^{20} - 116q^{21} - 199q^{22} - 16q^{23} - 115q^{24} - 147q^{25} - 89q^{26} - 125q^{27} - 176q^{28} - 10q^{29} - 42q^{30} - 68q^{31} + 29q^{32} - 7q^{33} - 190q^{34} + 16q^{35} + 33q^{36} - 54q^{37} + 80q^{38} - 9q^{39} - 218q^{40} - 78q^{41} - 72q^{42} - 212q^{43} - 119q^{44} - 136q^{45} - 360q^{46} - 92q^{47} - 167q^{48} - 301q^{49} - 169q^{50} - 202q^{51} - 327q^{52} - 170q^{53} - 155q^{54} - 242q^{55} - 240q^{56} - 116q^{57} - 286q^{58} - 48q^{59} - 166q^{60} - 218q^{61} - 96q^{62} - 36q^{63} - 243q^{64} - 78q^{65} - 31q^{66} - 200q^{67} - 2q^{68} + 2q^{69} - 208q^{70} + 48q^{71} + 29q^{72} - 114q^{73} + 6q^{74} - 39q^{75} - 260q^{76} - 68q^{77} + 41q^{78} - 336q^{79} - 138q^{80} + 13q^{81} - 122q^{82} - 80q^{83} - 100q^{84} - 192q^{85} - 76q^{86} - 126q^{87} - 267q^{88} - 130q^{89} - 174q^{90} - 276q^{91} - 172q^{92} - 226q^{93} - 376q^{94} - 276q^{95} - 207q^{96} - 286q^{97} - 239q^{98} - 159q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(429))$$

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
429.2.a $$\chi_{429}(1, \cdot)$$ 429.2.a.a 1 1
429.2.a.b 1
429.2.a.c 2
429.2.a.d 2
429.2.a.e 3
429.2.a.f 3
429.2.a.g 3
429.2.a.h 4
429.2.b $$\chi_{429}(298, \cdot)$$ 429.2.b.a 10 1
429.2.b.b 14
429.2.e $$\chi_{429}(428, \cdot)$$ 429.2.e.a 8 1
429.2.e.b 8
429.2.e.c 16
429.2.e.d 20
429.2.f $$\chi_{429}(131, \cdot)$$ 429.2.f.a 48 1
429.2.i $$\chi_{429}(100, \cdot)$$ 429.2.i.a 2 2
429.2.i.b 2
429.2.i.c 10
429.2.i.d 10
429.2.i.e 10
429.2.i.f 10
429.2.j $$\chi_{429}(122, \cdot)$$ 429.2.j.a 96 2
429.2.m $$\chi_{429}(109, \cdot)$$ 429.2.m.a 28 2
429.2.m.b 28
429.2.n $$\chi_{429}(157, \cdot)$$ 429.2.n.a 12 4
429.2.n.b 20
429.2.n.c 28
429.2.n.d 36
429.2.p $$\chi_{429}(230, \cdot)$$ 429.2.p.a 8 2
429.2.p.b 96
429.2.s $$\chi_{429}(166, \cdot)$$ 429.2.s.a 24 2
429.2.s.b 28
429.2.t $$\chi_{429}(296, \cdot)$$ 429.2.t.a 104 2
429.2.x $$\chi_{429}(248, \cdot)$$ 429.2.x.a 192 4
429.2.y $$\chi_{429}(116, \cdot)$$ 429.2.y.a 32 4
429.2.y.b 176
429.2.bb $$\chi_{429}(25, \cdot)$$ 429.2.bb.a 56 4
429.2.bb.b 56
429.2.bd $$\chi_{429}(76, \cdot)$$ 429.2.bd.a 56 4
429.2.bd.b 56
429.2.be $$\chi_{429}(89, \cdot)$$ 429.2.be.a 184 4
429.2.bg $$\chi_{429}(16, \cdot)$$ 429.2.bg.a 112 8
429.2.bg.b 112
429.2.bi $$\chi_{429}(5, \cdot)$$ 429.2.bi.a 416 8
429.2.bj $$\chi_{429}(73, \cdot)$$ 429.2.bj.a 112 8
429.2.bj.b 112
429.2.bm $$\chi_{429}(17, \cdot)$$ 429.2.bm.a 416 8
429.2.bn $$\chi_{429}(4, \cdot)$$ 429.2.bn.a 112 8
429.2.bn.b 112
429.2.bq $$\chi_{429}(29, \cdot)$$ 429.2.bq.a 416 8
429.2.bs $$\chi_{429}(7, \cdot)$$ 429.2.bs.a 224 16
429.2.bs.b 224
429.2.bv $$\chi_{429}(20, \cdot)$$ 429.2.bv.a 832 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(429))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(429)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(143))$$$$^{\oplus 2}$$