## Defining parameters

 Level: $$N$$ = $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$50$$ Sturm bound: $$34992$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(405))$$.

Total New Old
Modular forms 12096 8616 3480
Cusp forms 11232 8280 2952
Eisenstein series 864 336 528

## Trace form

 $$8280 q - 24 q^{2} - 36 q^{3} - 48 q^{4} - 45 q^{5} - 108 q^{6} - 46 q^{7} - 30 q^{8} - 36 q^{9} + O(q^{10})$$ $$8280 q - 24 q^{2} - 36 q^{3} - 48 q^{4} - 45 q^{5} - 108 q^{6} - 46 q^{7} - 30 q^{8} - 36 q^{9} - 83 q^{10} - 72 q^{11} - 36 q^{12} - 58 q^{13} - 42 q^{14} - 54 q^{15} - 56 q^{16} - 30 q^{17} + 108 q^{18} + 134 q^{19} + 399 q^{20} + 162 q^{21} + 232 q^{22} + 390 q^{23} + 180 q^{24} + 9 q^{25} - 42 q^{26} - 90 q^{27} - 338 q^{28} - 510 q^{29} - 270 q^{30} - 426 q^{31} - 1188 q^{32} - 414 q^{33} - 236 q^{34} - 201 q^{35} - 828 q^{36} - 166 q^{37} + 36 q^{38} - 36 q^{39} + 301 q^{40} + 972 q^{41} + 954 q^{42} + 596 q^{43} + 2574 q^{44} + 378 q^{45} + 1034 q^{46} + 1254 q^{47} + 954 q^{48} + 196 q^{49} - 27 q^{50} + 144 q^{51} - 322 q^{52} - 534 q^{53} - 288 q^{54} - 563 q^{55} - 2538 q^{56} - 468 q^{57} - 1278 q^{58} - 1896 q^{59} - 873 q^{60} - 1170 q^{61} - 3594 q^{62} - 1116 q^{63} - 1910 q^{64} - 1629 q^{65} - 3996 q^{66} - 184 q^{67} - 3996 q^{68} - 2196 q^{69} + 303 q^{70} - 726 q^{71} - 3492 q^{72} + 428 q^{73} + 162 q^{74} - 54 q^{75} + 1308 q^{76} + 678 q^{77} - 450 q^{78} + 890 q^{79} + 1572 q^{80} + 36 q^{81} + 580 q^{82} + 1074 q^{83} + 1818 q^{84} + 217 q^{85} + 1800 q^{86} + 1980 q^{87} - 144 q^{88} + 2736 q^{89} + 2457 q^{90} + 1360 q^{91} + 7134 q^{92} + 4356 q^{93} - 146 q^{94} + 2307 q^{95} + 6498 q^{96} + 80 q^{97} + 5370 q^{98} + 3060 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(405))$$

We only show spaces with odd 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
405.3.c $$\chi_{405}(161, \cdot)$$ 405.3.c.a 16 1
405.3.c.b 16
405.3.d $$\chi_{405}(404, \cdot)$$ 405.3.d.a 20 1
405.3.d.b 24
405.3.g $$\chi_{405}(82, \cdot)$$ 405.3.g.a 4 2
405.3.g.b 4
405.3.g.c 8
405.3.g.d 8
405.3.g.e 8
405.3.g.f 16
405.3.g.g 20
405.3.g.h 20
405.3.h $$\chi_{405}(134, \cdot)$$ 405.3.h.a 2 2
405.3.h.b 2
405.3.h.c 4
405.3.h.d 4
405.3.h.e 4
405.3.h.f 4
405.3.h.g 4
405.3.h.h 4
405.3.h.i 8
405.3.h.j 8
405.3.h.k 48
405.3.i $$\chi_{405}(26, \cdot)$$ 405.3.i.a 4 2
405.3.i.b 4
405.3.i.c 8
405.3.i.d 8
405.3.i.e 8
405.3.i.f 32
405.3.l $$\chi_{405}(28, \cdot)$$ 405.3.l.a 4 4
405.3.l.b 4
405.3.l.c 4
405.3.l.d 4
405.3.l.e 8
405.3.l.f 8
405.3.l.g 8
405.3.l.h 8
405.3.l.i 8
405.3.l.j 16
405.3.l.k 16
405.3.l.l 16
405.3.l.m 16
405.3.l.n 32
405.3.l.o 32
405.3.n $$\chi_{405}(44, \cdot)$$ 405.3.n.a 204 6
405.3.o $$\chi_{405}(71, \cdot)$$ 405.3.o.a 144 6
405.3.s $$\chi_{405}(37, \cdot)$$ 405.3.s.a 408 12
405.3.u $$\chi_{405}(11, \cdot)$$ 405.3.u.a 1296 18
405.3.v $$\chi_{405}(14, \cdot)$$ 405.3.v.a 1908 18
405.3.w $$\chi_{405}(7, \cdot)$$ 405.3.w.a 3816 36

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(405))$$ into lower level spaces

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