# Properties

 Label 405.2 Level 405 Weight 2 Dimension 4056 Nonzero newspaces 12 Newform subspaces 50 Sturm bound 23328 Trace bound 1

## 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

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

## 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 available 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

$$S_{2}^{\mathrm{old}}(\Gamma_1(405)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 1}$$