## Defining parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$48$$ Sturm bound: $$14400$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(350))$$.

Total New Old
Modular forms 3936 1058 2878
Cusp forms 3265 1058 2207
Eisenstein series 671 0 671

## Trace form

 $$1058q + 2q^{2} + 10q^{3} + 4q^{4} + 10q^{5} + 14q^{6} + 16q^{7} + 2q^{8} + 40q^{9} + O(q^{10})$$ $$1058q + 2q^{2} + 10q^{3} + 4q^{4} + 10q^{5} + 14q^{6} + 16q^{7} + 2q^{8} + 40q^{9} + 10q^{10} + 36q^{11} + 10q^{12} + 38q^{13} + 14q^{14} + 40q^{15} + 4q^{16} + 20q^{17} - 12q^{18} - 18q^{19} + 18q^{21} - 44q^{22} - 8q^{23} - 26q^{24} - 94q^{25} - 42q^{26} - 140q^{27} - 52q^{28} - 92q^{29} - 112q^{30} - 84q^{31} - 8q^{32} - 176q^{33} - 98q^{34} - 52q^{35} - 56q^{36} - 38q^{37} - 38q^{38} - 144q^{39} + 10q^{40} - 4q^{41} - 82q^{42} - 36q^{43} - 12q^{44} - 102q^{45} + 24q^{46} + 4q^{47} + 10q^{48} + 90q^{49} + 50q^{50} + 84q^{51} + 38q^{52} + 78q^{53} + 116q^{54} + 16q^{55} + 14q^{56} - 20q^{57} + 96q^{58} - 42q^{59} - 70q^{61} - 20q^{62} - 136q^{63} + 4q^{64} - 142q^{65} + 24q^{66} - 152q^{67} - 60q^{68} - 184q^{69} - 40q^{70} - 40q^{71} + 38q^{72} - 128q^{73} - 88q^{74} - 216q^{75} - 18q^{76} - 124q^{77} - 40q^{78} - 104q^{79} + 10q^{80} + 48q^{81} - 40q^{82} - 170q^{83} - 2q^{84} - 78q^{85} - 44q^{86} - 100q^{87} - 12q^{88} - 186q^{89} - 86q^{90} - 66q^{91} - 64q^{92} - 64q^{93} - 36q^{94} - 72q^{95} - 34q^{96} - 44q^{97} - 190q^{98} - 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(350))$$

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
350.2.a $$\chi_{350}(1, \cdot)$$ 350.2.a.a 1 1
350.2.a.b 1
350.2.a.c 1
350.2.a.d 1
350.2.a.e 1
350.2.a.f 1
350.2.a.g 2
350.2.a.h 2
350.2.c $$\chi_{350}(99, \cdot)$$ 350.2.c.a 2 1
350.2.c.b 2
350.2.c.c 2
350.2.c.d 2
350.2.e $$\chi_{350}(51, \cdot)$$ 350.2.e.a 2 2
350.2.e.b 2
350.2.e.c 2
350.2.e.d 2
350.2.e.e 2
350.2.e.f 2
350.2.e.g 2
350.2.e.h 2
350.2.e.i 2
350.2.e.j 2
350.2.e.k 2
350.2.e.l 2
350.2.g $$\chi_{350}(293, \cdot)$$ 350.2.g.a 8 2
350.2.g.b 16
350.2.h $$\chi_{350}(71, \cdot)$$ 350.2.h.a 8 4
350.2.h.b 12
350.2.h.c 16
350.2.h.d 20
350.2.j $$\chi_{350}(149, \cdot)$$ 350.2.j.a 4 2
350.2.j.b 4
350.2.j.c 4
350.2.j.d 4
350.2.j.e 4
350.2.j.f 4
350.2.m $$\chi_{350}(29, \cdot)$$ 350.2.m.a 24 4
350.2.m.b 40
350.2.o $$\chi_{350}(143, \cdot)$$ 350.2.o.a 8 4
350.2.o.b 8
350.2.o.c 16
350.2.o.d 16
350.2.q $$\chi_{350}(11, \cdot)$$ 350.2.q.a 8 8
350.2.q.b 72
350.2.q.c 80
350.2.r $$\chi_{350}(13, \cdot)$$ 350.2.r.a 160 8
350.2.u $$\chi_{350}(9, \cdot)$$ 350.2.u.a 160 8
350.2.x $$\chi_{350}(3, \cdot)$$ 350.2.x.a 320 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(350))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(350)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(175))$$$$^{\oplus 2}$$