## Defining parameters

 Level: $$N$$ = $$418 = 2 \cdot 11 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$50$$ Sturm bound: $$21600$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(418))$$.

Total New Old
Modular forms 5760 1763 3997
Cusp forms 5041 1763 3278
Eisenstein series 719 0 719

## Trace form

 $$1763 q + 3 q^{2} + 12 q^{3} + 3 q^{4} + 18 q^{5} + 2 q^{6} + 4 q^{7} + 3 q^{8} - q^{9} + O(q^{10})$$ $$1763 q + 3 q^{2} + 12 q^{3} + 3 q^{4} + 18 q^{5} + 2 q^{6} + 4 q^{7} + 3 q^{8} - q^{9} - 2 q^{10} + 3 q^{11} - 20 q^{12} - 26 q^{13} - 32 q^{14} - 60 q^{15} + 3 q^{16} - 22 q^{17} - 25 q^{18} - 60 q^{19} - 18 q^{20} - 28 q^{21} - 4 q^{22} - 4 q^{23} + 2 q^{24} - 59 q^{25} - 34 q^{26} - 36 q^{27} - 8 q^{28} - 26 q^{29} + 12 q^{30} - 7 q^{32} - 72 q^{33} + 14 q^{34} - 48 q^{35} + 29 q^{36} + 38 q^{37} + 9 q^{38} - 40 q^{39} - 2 q^{40} + 10 q^{41} + 36 q^{42} - 52 q^{43} - 16 q^{44} - 82 q^{45} - 40 q^{46} - 80 q^{47} - 6 q^{48} - 69 q^{49} - 91 q^{50} - 112 q^{51} - 10 q^{52} - 82 q^{53} - 68 q^{54} - 74 q^{55} - 48 q^{56} - 109 q^{57} - 62 q^{58} - 130 q^{59} - 40 q^{60} - 110 q^{61} - 72 q^{62} - 120 q^{63} - 9 q^{64} - 164 q^{65} - 100 q^{66} - 132 q^{67} - 4 q^{68} - 40 q^{69} - 8 q^{70} - 88 q^{71} - 19 q^{72} + 36 q^{73} + 74 q^{74} + 58 q^{75} + 29 q^{76} - 12 q^{77} + 16 q^{78} - 16 q^{79} - 2 q^{80} - 105 q^{81} - 16 q^{82} - 22 q^{83} - 92 q^{84} - 40 q^{85} - 70 q^{86} - 200 q^{87} + 3 q^{88} - 34 q^{89} - 126 q^{90} - 56 q^{91} - 60 q^{92} - 176 q^{93} - 120 q^{94} - 56 q^{95} + 12 q^{96} - 76 q^{97} - 63 q^{98} - 168 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(418))$$

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
418.2.a $$\chi_{418}(1, \cdot)$$ 418.2.a.a 1 1
418.2.a.b 1
418.2.a.c 1
418.2.a.d 2
418.2.a.e 2
418.2.a.f 2
418.2.a.g 3
418.2.a.h 3
418.2.b $$\chi_{418}(417, \cdot)$$ 418.2.b.a 2 1
418.2.b.b 2
418.2.b.c 8
418.2.b.d 8
418.2.e $$\chi_{418}(45, \cdot)$$ 418.2.e.a 2 2
418.2.e.b 2
418.2.e.c 2
418.2.e.d 2
418.2.e.e 2
418.2.e.f 2
418.2.e.g 4
418.2.e.h 6
418.2.e.i 6
418.2.f $$\chi_{418}(115, \cdot)$$ 418.2.f.a 4 4
418.2.f.b 4
418.2.f.c 4
418.2.f.d 4
418.2.f.e 4
418.2.f.f 16
418.2.f.g 16
418.2.f.h 20
418.2.h $$\chi_{418}(65, \cdot)$$ 418.2.h.a 20 2
418.2.h.b 20
418.2.j $$\chi_{418}(23, \cdot)$$ 418.2.j.a 24 6
418.2.j.b 24
418.2.j.c 30
418.2.j.d 30
418.2.m $$\chi_{418}(151, \cdot)$$ 418.2.m.a 40 4
418.2.m.b 40
418.2.n $$\chi_{418}(49, \cdot)$$ 418.2.n.a 8 8
418.2.n.b 8
418.2.n.c 8
418.2.n.d 64
418.2.n.e 72
418.2.q $$\chi_{418}(21, \cdot)$$ 418.2.q.a 60 6
418.2.q.b 60
418.2.s $$\chi_{418}(107, \cdot)$$ 418.2.s.a 80 8
418.2.s.b 80
418.2.u $$\chi_{418}(5, \cdot)$$ 418.2.u.a 216 24
418.2.u.b 264
418.2.v $$\chi_{418}(13, \cdot)$$ 418.2.v.a 240 24
418.2.v.b 240

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(418))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(418)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(209))$$$$^{\oplus 2}$$