## Defining parameters

 Level: $$N$$ = $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Newform subspaces: $$27$$ Sturm bound: $$3072$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(105))$$.

Total New Old
Modular forms 1248 784 464
Cusp forms 1056 728 328
Eisenstein series 192 56 136

## Trace form

 $$728q - 8q^{2} - 6q^{3} + 36q^{4} + 12q^{5} + 48q^{6} + 104q^{7} - 36q^{8} - 54q^{9} + O(q^{10})$$ $$728q - 8q^{2} - 6q^{3} + 36q^{4} + 12q^{5} + 48q^{6} + 104q^{7} - 36q^{8} - 54q^{9} + 120q^{10} + 112q^{11} - 36q^{12} - 184q^{13} + 360q^{14} - 174q^{15} - 404q^{16} - 56q^{17} - 144q^{18} - 628q^{19} - 172q^{20} - 606q^{21} + 128q^{22} + 912q^{23} + 1044q^{24} + 1574q^{25} + 1948q^{26} - 408q^{27} + 2268q^{28} - 472q^{29} - 1722q^{30} - 868q^{31} - 1828q^{32} - 306q^{33} - 1576q^{34} - 2012q^{35} - 2244q^{36} - 2364q^{37} - 4748q^{38} - 2028q^{39} - 1712q^{40} + 728q^{41} - 3360q^{42} - 1432q^{43} - 448q^{44} + 321q^{45} + 2456q^{46} + 880q^{47} + 6252q^{48} + 3832q^{49} + 3700q^{50} + 7002q^{51} + 10584q^{52} + 3976q^{53} + 8736q^{54} + 7812q^{55} + 9468q^{56} + 2460q^{57} + 4904q^{58} + 3616q^{59} - 1704q^{60} - 8124q^{61} - 7056q^{62} - 6618q^{63} - 22284q^{64} - 8936q^{65} - 12588q^{66} - 14204q^{67} - 14920q^{68} - 3600q^{69} - 19800q^{70} - 4072q^{71} + 7896q^{72} + 3644q^{73} - 724q^{74} + 6645q^{75} + 11104q^{76} + 216q^{77} + 1320q^{78} + 8068q^{79} + 3752q^{80} - 9546q^{81} - 1944q^{82} - 288q^{83} - 7476q^{84} + 3516q^{85} + 388q^{86} - 3396q^{87} + 11520q^{88} + 5256q^{89} - 9876q^{90} - 3248q^{91} + 7296q^{92} - 4986q^{93} - 7864q^{94} - 7952q^{95} + 4896q^{96} - 5592q^{97} + 12932q^{98} + 13956q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(105))$$

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
105.4.a $$\chi_{105}(1, \cdot)$$ 105.4.a.a 1 1
105.4.a.b 1
105.4.a.c 2
105.4.a.d 2
105.4.a.e 2
105.4.a.f 2
105.4.a.g 2
105.4.b $$\chi_{105}(41, \cdot)$$ 105.4.b.a 16 1
105.4.b.b 16
105.4.d $$\chi_{105}(64, \cdot)$$ 105.4.d.a 6 1
105.4.d.b 10
105.4.g $$\chi_{105}(104, \cdot)$$ 105.4.g.a 4 1
105.4.g.b 40
105.4.i $$\chi_{105}(16, \cdot)$$ 105.4.i.a 2 2
105.4.i.b 4
105.4.i.c 6
105.4.i.d 10
105.4.i.e 10
105.4.j $$\chi_{105}(8, \cdot)$$ 105.4.j.a 72 2
105.4.m $$\chi_{105}(13, \cdot)$$ 105.4.m.a 48 2
105.4.p $$\chi_{105}(59, \cdot)$$ 105.4.p.a 88 2
105.4.q $$\chi_{105}(4, \cdot)$$ 105.4.q.a 4 2
105.4.q.b 44
105.4.s $$\chi_{105}(26, \cdot)$$ 105.4.s.a 32 2
105.4.s.b 32
105.4.u $$\chi_{105}(52, \cdot)$$ 105.4.u.a 96 4
105.4.x $$\chi_{105}(2, \cdot)$$ 105.4.x.a 176 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(105))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(105)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$