# Properties

 Label 330.2 Level 330 Weight 2 Dimension 645 Nonzero newspaces 12 Newforms 31 Sturm bound 11520 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$330 = 2 \cdot 3 \cdot 5 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newforms: $$31$$ Sturm bound: $$11520$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(330))$$.

Total New Old
Modular forms 3200 645 2555
Cusp forms 2561 645 1916
Eisenstein series 639 0 639

## Trace form

 $$645q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 15q^{6} + 48q^{7} + q^{8} + 41q^{9} + O(q^{10})$$ $$645q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 15q^{6} + 48q^{7} + q^{8} + 41q^{9} + 13q^{10} + 25q^{11} + 9q^{12} + 22q^{13} + 16q^{14} + 11q^{15} + q^{16} + 50q^{17} - 5q^{18} + 48q^{19} - 7q^{20} + 32q^{21} - 7q^{22} + 16q^{23} - 5q^{24} + 57q^{25} + 14q^{26} - 55q^{27} + 8q^{28} + 22q^{29} - q^{30} + 40q^{31} + q^{32} - 97q^{33} - 14q^{34} - 4q^{35} - 9q^{36} - 90q^{37} - 28q^{38} - 62q^{39} - 23q^{40} + 2q^{41} - 84q^{42} - 124q^{43} - 47q^{44} + 9q^{45} - 152q^{46} - 128q^{47} - 11q^{48} - 247q^{49} - 135q^{50} - 56q^{51} - 98q^{52} - 162q^{53} - 15q^{54} - 235q^{55} - 8q^{56} - 130q^{57} - 162q^{58} - 140q^{59} - 47q^{60} - 122q^{61} - 56q^{62} - 52q^{63} + q^{64} - 106q^{65} + 17q^{66} - 68q^{67} - 30q^{68} - 28q^{69} + 8q^{70} + 40q^{71} + 57q^{72} + 122q^{73} + 38q^{74} + 27q^{75} - 4q^{76} + 72q^{77} + 70q^{78} + 24q^{79} + 29q^{80} + 17q^{81} - 2q^{82} + 12q^{83} - 8q^{84} + 30q^{85} + 52q^{86} - 58q^{87} + 33q^{88} - 46q^{89} - 33q^{90} - 304q^{91} - 24q^{92} - 124q^{93} - 96q^{94} - 128q^{95} - 11q^{96} - 306q^{97} - 39q^{98} - 267q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(330))$$

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
330.2.a $$\chi_{330}(1, \cdot)$$ 330.2.a.a 1 1
330.2.a.b 1
330.2.a.c 1
330.2.a.d 1
330.2.a.e 1
330.2.c $$\chi_{330}(199, \cdot)$$ 330.2.c.a 4 1
330.2.c.b 4
330.2.d $$\chi_{330}(131, \cdot)$$ 330.2.d.a 8 1
330.2.d.b 8
330.2.f $$\chi_{330}(329, \cdot)$$ 330.2.f.a 24 1
330.2.j $$\chi_{330}(23, \cdot)$$ 330.2.j.a 20 2
330.2.j.b 20
330.2.l $$\chi_{330}(43, \cdot)$$ 330.2.l.a 4 2
330.2.l.b 4
330.2.l.c 8
330.2.l.d 8
330.2.m $$\chi_{330}(31, \cdot)$$ 330.2.m.a 4 4
330.2.m.b 4
330.2.m.c 4
330.2.m.d 4
330.2.m.e 8
330.2.m.f 8
330.2.p $$\chi_{330}(29, \cdot)$$ 330.2.p.a 96 4
330.2.r $$\chi_{330}(41, \cdot)$$ 330.2.r.a 32 4
330.2.r.b 32
330.2.s $$\chi_{330}(49, \cdot)$$ 330.2.s.a 8 4
330.2.s.b 8
330.2.s.c 32
330.2.u $$\chi_{330}(7, \cdot)$$ 330.2.u.a 48 8
330.2.u.b 48
330.2.w $$\chi_{330}(47, \cdot)$$ 330.2.w.a 192 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(330))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(330)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(66))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(110))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(165))$$$$^{\oplus 2}$$