## Defining parameters

 Level: $$N$$ = $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$50$$ Sturm bound: $$23328$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(405))$$.

Total New Old
Modular forms 6264 4392 1872
Cusp forms 5401 4056 1345
Eisenstein series 863 336 527

## Trace form

 $$4056q - 24q^{2} - 36q^{3} - 36q^{4} - 33q^{5} - 108q^{6} - 34q^{7} - 36q^{9} + O(q^{10})$$ $$4056q - 24q^{2} - 36q^{3} - 36q^{4} - 33q^{5} - 108q^{6} - 34q^{7} - 36q^{9} - 77q^{10} - 54q^{11} - 36q^{12} - 22q^{13} + 18q^{14} - 54q^{15} - 116q^{16} + 6q^{17} - 54q^{18} - 70q^{19} - 69q^{20} - 162q^{21} - 50q^{22} - 90q^{23} - 144q^{24} - 75q^{25} - 246q^{26} - 90q^{27} - 122q^{28} - 78q^{29} - 108q^{30} - 138q^{31} - 168q^{32} - 90q^{33} - 86q^{34} - 81q^{35} - 180q^{36} - 58q^{37} - 18q^{38} - 36q^{39} - 155q^{40} - 114q^{41} - 126q^{42} - 82q^{43} - 282q^{44} - 108q^{45} - 310q^{46} - 210q^{47} - 234q^{48} - 128q^{49} - 291q^{50} - 234q^{51} - 190q^{52} - 234q^{53} - 288q^{54} - 155q^{55} - 486q^{56} - 144q^{57} - 126q^{58} - 174q^{59} - 171q^{60} - 150q^{61} - 234q^{62} - 144q^{63} - 116q^{64} - 45q^{65} - 108q^{66} - 58q^{67} + 120q^{68} + 72q^{69} - 141q^{70} + 138q^{71} + 396q^{72} - 4q^{73} + 258q^{74} - 54q^{75} - 234q^{76} + 210q^{77} + 198q^{78} - 106q^{79} + 102q^{80} + 36q^{81} - 128q^{82} + 90q^{83} + 414q^{84} - 161q^{85} + 114q^{86} + 252q^{87} - 234q^{88} - 24q^{89} + 27q^{90} - 260q^{91} - 90q^{92} - 72q^{93} - 266q^{94} - 219q^{95} - 90q^{96} - 214q^{97} - 528q^{98} - 180q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(405))$$

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
405.2.a $$\chi_{405}(1, \cdot)$$ 405.2.a.a 1 1
405.2.a.b 1
405.2.a.c 1
405.2.a.d 1
405.2.a.e 1
405.2.a.f 1
405.2.a.g 2
405.2.a.h 2
405.2.a.i 3
405.2.a.j 3
405.2.b $$\chi_{405}(244, \cdot)$$ 405.2.b.a 4 1
405.2.b.b 4
405.2.b.c 4
405.2.b.d 4
405.2.b.e 4
405.2.e $$\chi_{405}(136, \cdot)$$ 405.2.e.a 2 2
405.2.e.b 2
405.2.e.c 2
405.2.e.d 2
405.2.e.e 2
405.2.e.f 2
405.2.e.g 2
405.2.e.h 2
405.2.e.i 4
405.2.e.j 4
405.2.e.k 4
405.2.e.l 4
405.2.f $$\chi_{405}(242, \cdot)$$ 405.2.f.a 16 2
405.2.f.b 24
405.2.j $$\chi_{405}(109, \cdot)$$ 405.2.j.a 4 2
405.2.j.b 4
405.2.j.c 4
405.2.j.d 4
405.2.j.e 4
405.2.j.f 8
405.2.j.g 8
405.2.j.h 8
405.2.k $$\chi_{405}(46, \cdot)$$ 405.2.k.a 30 6
405.2.k.b 42
405.2.m $$\chi_{405}(53, \cdot)$$ 405.2.m.a 8 4
405.2.m.b 16
405.2.m.c 16
405.2.m.d 24
405.2.m.e 24
405.2.p $$\chi_{405}(19, \cdot)$$ 405.2.p.a 96 6
405.2.q $$\chi_{405}(16, \cdot)$$ 405.2.q.a 306 18
405.2.q.b 342
405.2.r $$\chi_{405}(8, \cdot)$$ 405.2.r.a 192 12
405.2.t $$\chi_{405}(4, \cdot)$$ 405.2.t.a 936 18
405.2.x $$\chi_{405}(2, \cdot)$$ 405.2.x.a 1872 36

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(405))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(405)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 2}$$