## Defining parameters

 Level: $$N$$ = $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$46656$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(405))$$.

Total New Old
Modular forms 17928 12840 5088
Cusp forms 17064 12504 4560
Eisenstein series 864 336 528

## Trace form

 $$12504q - 24q^{2} - 36q^{3} - 24q^{4} - 24q^{5} - 108q^{6} - 76q^{7} - 162q^{8} - 36q^{9} + O(q^{10})$$ $$12504q - 24q^{2} - 36q^{3} - 24q^{4} - 24q^{5} - 108q^{6} - 76q^{7} - 162q^{8} - 36q^{9} - 107q^{10} + 54q^{11} - 36q^{12} + 68q^{13} + 90q^{14} - 54q^{15} - 464q^{16} - 426q^{17} - 864q^{18} - 958q^{19} - 1203q^{20} - 324q^{21} + 316q^{22} + 1548q^{23} + 2124q^{24} + 678q^{25} + 6558q^{26} + 1368q^{27} + 2974q^{28} + 2028q^{29} + 702q^{30} - 228q^{31} - 2508q^{32} - 900q^{33} - 3392q^{34} - 4185q^{35} - 4716q^{36} - 3406q^{37} - 4824q^{38} - 36q^{39} - 341q^{40} - 5694q^{41} - 7146q^{42} - 922q^{43} - 4494q^{44} - 1080q^{45} + 2234q^{46} + 5424q^{47} + 4518q^{48} + 4714q^{49} + 10095q^{50} + 5814q^{51} + 7250q^{52} + 9486q^{53} + 13320q^{54} + 2575q^{55} + 17118q^{56} + 4392q^{57} + 3582q^{58} + 5370q^{59} + 1233q^{60} - 48q^{61} - 8010q^{62} - 4032q^{63} - 12038q^{64} - 5526q^{65} + 3780q^{66} - 12946q^{67} - 18096q^{68} + 720q^{69} - 7119q^{70} - 4830q^{71} - 3492q^{72} - 3730q^{73} - 16554q^{74} - 54q^{75} + 2436q^{76} - 16584q^{77} - 21834q^{78} + 6296q^{79} - 10482q^{80} - 11628q^{81} + 5788q^{82} + 1908q^{83} - 34794q^{84} + 8317q^{85} + 9420q^{86} - 10980q^{87} + 7848q^{88} - 11742q^{89} - 11151q^{90} - 11294q^{91} - 19566q^{92} - 7632q^{93} - 16802q^{94} + 1005q^{95} + 34362q^{96} - 8014q^{97} + 33438q^{98} + 26388q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(405))$$

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
405.4.a $$\chi_{405}(1, \cdot)$$ 405.4.a.a 1 1
405.4.a.b 1
405.4.a.c 2
405.4.a.d 2
405.4.a.e 2
405.4.a.f 2
405.4.a.g 3
405.4.a.h 3
405.4.a.i 3
405.4.a.j 3
405.4.a.k 6
405.4.a.l 6
405.4.a.m 7
405.4.a.n 7
405.4.b $$\chi_{405}(244, \cdot)$$ 405.4.b.a 4 1
405.4.b.b 8
405.4.b.c 8
405.4.b.d 16
405.4.b.e 16
405.4.b.f 16
405.4.e $$\chi_{405}(136, \cdot)$$ 405.4.e.a 2 2
405.4.e.b 2
405.4.e.c 2
405.4.e.d 2
405.4.e.e 2
405.4.e.f 2
405.4.e.g 2
405.4.e.h 2
405.4.e.i 2
405.4.e.j 2
405.4.e.k 2
405.4.e.l 2
405.4.e.m 2
405.4.e.n 2
405.4.e.o 4
405.4.e.p 4
405.4.e.q 6
405.4.e.r 6
405.4.e.s 6
405.4.e.t 6
405.4.e.u 6
405.4.e.v 6
405.4.e.w 12
405.4.e.x 12
405.4.f $$\chi_{405}(242, \cdot)$$ n/a 136 2
405.4.j $$\chi_{405}(109, \cdot)$$ n/a 140 2
405.4.k $$\chi_{405}(46, \cdot)$$ n/a 216 6
405.4.m $$\chi_{405}(53, \cdot)$$ n/a 280 4
405.4.p $$\chi_{405}(19, \cdot)$$ n/a 312 6
405.4.q $$\chi_{405}(16, \cdot)$$ n/a 1944 18
405.4.r $$\chi_{405}(8, \cdot)$$ n/a 624 12
405.4.t $$\chi_{405}(4, \cdot)$$ n/a 2880 18
405.4.x $$\chi_{405}(2, \cdot)$$ n/a 5760 36

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(405))$$ into lower level spaces

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