## Defining parameters

 Level: $$N$$ = $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$20$$ Newform subspaces: $$70$$ Sturm bound: $$16128$$ Trace bound: $$18$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(390))$$.

Total New Old
Modular forms 4416 909 3507
Cusp forms 3649 909 2740
Eisenstein series 767 0 767

## Trace form

 $$909q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 5q^{6} + 24q^{7} + 13q^{8} + 9q^{9} + O(q^{10})$$ $$909q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 5q^{6} + 24q^{7} + 13q^{8} + 9q^{9} + 23q^{10} + 28q^{11} - 3q^{12} + 81q^{13} + 24q^{14} + 5q^{15} + 17q^{16} + 30q^{17} - 3q^{18} + 100q^{19} - q^{20} + 48q^{21} - 4q^{22} + 24q^{23} + 5q^{24} + 17q^{25} + q^{26} + 77q^{27} + 8q^{28} + 42q^{29} + 5q^{30} + 48q^{31} + q^{32} - 28q^{33} - 14q^{34} - 47q^{36} - 78q^{37} - 28q^{38} - 115q^{39} - 23q^{40} - 26q^{41} - 96q^{42} - 20q^{43} - 4q^{44} - 45q^{45} - 72q^{46} - 11q^{48} - 71q^{49} - 33q^{50} - 54q^{51} - 27q^{52} - 90q^{53} - 67q^{54} - 188q^{55} - 56q^{56} - 92q^{57} - 110q^{58} - 116q^{59} - 59q^{60} - 38q^{61} - 160q^{62} - 200q^{63} - 11q^{64} - 281q^{65} - 132q^{66} - 204q^{67} - 42q^{68} - 248q^{69} - 208q^{70} - 216q^{71} - 31q^{72} - 166q^{73} - 102q^{74} - 275q^{75} - 28q^{76} - 96q^{77} - 35q^{78} - 9q^{80} - 199q^{81} + 158q^{82} + 36q^{83} + 48q^{84} + 120q^{85} + 28q^{86} - 10q^{87} - 4q^{88} + 74q^{89} + q^{90} + 88q^{91} + 72q^{92} + 16q^{93} + 224q^{94} + 68q^{95} - 11q^{96} + 162q^{97} + 57q^{98} - 76q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(390))$$

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
390.2.a $$\chi_{390}(1, \cdot)$$ 390.2.a.a 1 1
390.2.a.b 1
390.2.a.c 1
390.2.a.d 1
390.2.a.e 1
390.2.a.f 1
390.2.a.g 1
390.2.a.h 2
390.2.b $$\chi_{390}(181, \cdot)$$ 390.2.b.a 2 1
390.2.b.b 2
390.2.b.c 4
390.2.b.d 4
390.2.e $$\chi_{390}(79, \cdot)$$ 390.2.e.a 2 1
390.2.e.b 2
390.2.e.c 2
390.2.e.d 2
390.2.e.e 4
390.2.f $$\chi_{390}(259, \cdot)$$ 390.2.f.a 6 1
390.2.f.b 6
390.2.i $$\chi_{390}(61, \cdot)$$ 390.2.i.a 2 2
390.2.i.b 2
390.2.i.c 2
390.2.i.d 2
390.2.i.e 2
390.2.i.f 2
390.2.i.g 4
390.2.j $$\chi_{390}(73, \cdot)$$ 390.2.j.a 12 2
390.2.j.b 16
390.2.l $$\chi_{390}(53, \cdot)$$ 390.2.l.a 4 2
390.2.l.b 4
390.2.l.c 20
390.2.l.d 20
390.2.n $$\chi_{390}(239, \cdot)$$ 390.2.n.a 56 2
390.2.p $$\chi_{390}(161, \cdot)$$ 390.2.p.a 4 2
390.2.p.b 4
390.2.p.c 4
390.2.p.d 4
390.2.p.e 4
390.2.p.f 4
390.2.p.g 8
390.2.s $$\chi_{390}(77, \cdot)$$ 390.2.s.a 8 2
390.2.s.b 24
390.2.s.c 24
390.2.t $$\chi_{390}(307, \cdot)$$ 390.2.t.a 12 2
390.2.t.b 16
390.2.x $$\chi_{390}(49, \cdot)$$ 390.2.x.a 12 2
390.2.x.b 12
390.2.y $$\chi_{390}(139, \cdot)$$ 390.2.y.a 4 2
390.2.y.b 4
390.2.y.c 4
390.2.y.d 4
390.2.y.e 4
390.2.y.f 4
390.2.y.g 8
390.2.bb $$\chi_{390}(121, \cdot)$$ 390.2.bb.a 4 2
390.2.bb.b 4
390.2.bb.c 8
390.2.bd $$\chi_{390}(7, \cdot)$$ 390.2.bd.a 8 4
390.2.bd.b 16
390.2.bd.c 32
390.2.be $$\chi_{390}(17, \cdot)$$ 390.2.be.a 16 4
390.2.be.b 96
390.2.bh $$\chi_{390}(11, \cdot)$$ 390.2.bh.a 8 4
390.2.bh.b 32
390.2.bh.c 40
390.2.bj $$\chi_{390}(59, \cdot)$$ 390.2.bj.a 112 4
390.2.bl $$\chi_{390}(107, \cdot)$$ 390.2.bl.a 112 4
390.2.bn $$\chi_{390}(67, \cdot)$$ 390.2.bn.a 8 4
390.2.bn.b 16
390.2.bn.c 32

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(390))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(390)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(130))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(195))$$$$^{\oplus 2}$$