## Defining parameters

 Level: $$N$$ = $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$279936$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1620))$$.

Total New Old
Modular forms 72144 29904 42240
Cusp forms 67825 29232 38593
Eisenstein series 4319 672 3647

## Trace form

 $$29232q - 24q^{2} - 40q^{4} - 69q^{5} - 108q^{6} + 6q^{7} - 30q^{8} - 72q^{9} + O(q^{10})$$ $$29232q - 24q^{2} - 40q^{4} - 69q^{5} - 108q^{6} + 6q^{7} - 30q^{8} - 72q^{9} - 91q^{10} + 6q^{11} - 36q^{12} - 86q^{13} - 54q^{14} - 144q^{16} - 84q^{17} - 36q^{18} - 36q^{19} - 57q^{20} - 270q^{21} - 56q^{22} - 114q^{23} - 36q^{24} - 147q^{25} - 42q^{26} - 54q^{27} - 50q^{28} - 174q^{29} - 54q^{30} - 54q^{31} + 36q^{32} - 126q^{33} + 24q^{34} - 84q^{35} - 108q^{36} - 188q^{37} + 24q^{38} - 7q^{40} - 174q^{41} + 54q^{42} + 6q^{43} + 270q^{44} - 54q^{45} + 2q^{46} + 138q^{47} + 162q^{48} - 42q^{49} + 159q^{50} + 126q^{51} + 102q^{52} + 192q^{53} + 216q^{54} + 66q^{55} + 390q^{56} + 36q^{57} + 118q^{58} + 234q^{59} + 63q^{60} - 126q^{61} + 366q^{62} + 108q^{63} + 122q^{64} + 75q^{65} + 36q^{66} + 18q^{67} + 228q^{68} + 108q^{69} + 41q^{70} + 120q^{71} - 36q^{72} - 62q^{73} + 114q^{74} + 44q^{76} + 234q^{77} - 90q^{78} - 42q^{79} + 132q^{80} - 72q^{81} - 12q^{82} + 174q^{83} + 18q^{84} - 20q^{85} + 96q^{86} + 288q^{87} - 56q^{88} + 282q^{89} - 117q^{90} + 126q^{91} - 210q^{92} + 324q^{93} - 86q^{94} + 252q^{95} - 342q^{96} + 70q^{97} - 486q^{98} + 180q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1620))$$

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
1620.2.a $$\chi_{1620}(1, \cdot)$$ 1620.2.a.a 1 1
1620.2.a.b 1
1620.2.a.c 1
1620.2.a.d 1
1620.2.a.e 1
1620.2.a.f 1
1620.2.a.g 2
1620.2.a.h 2
1620.2.a.i 3
1620.2.a.j 3
1620.2.d $$\chi_{1620}(649, \cdot)$$ 1620.2.d.a 2 1
1620.2.d.b 2
1620.2.d.c 6
1620.2.d.d 6
1620.2.d.e 8
1620.2.e $$\chi_{1620}(971, \cdot)$$ 1620.2.e.a 48 1
1620.2.e.b 48
1620.2.h $$\chi_{1620}(1619, \cdot)$$ n/a 136 1
1620.2.i $$\chi_{1620}(541, \cdot)$$ 1620.2.i.a 2 2
1620.2.i.b 2
1620.2.i.c 2
1620.2.i.d 2
1620.2.i.e 2
1620.2.i.f 2
1620.2.i.g 2
1620.2.i.h 2
1620.2.i.i 2
1620.2.i.j 2
1620.2.i.k 2
1620.2.i.l 2
1620.2.i.m 4
1620.2.i.n 4
1620.2.j $$\chi_{1620}(1133, \cdot)$$ 1620.2.j.a 24 2
1620.2.j.b 24
1620.2.k $$\chi_{1620}(163, \cdot)$$ n/a 272 2
1620.2.n $$\chi_{1620}(539, \cdot)$$ n/a 280 2
1620.2.q $$\chi_{1620}(431, \cdot)$$ n/a 192 2
1620.2.r $$\chi_{1620}(109, \cdot)$$ 1620.2.r.a 4 2
1620.2.r.b 4
1620.2.r.c 4
1620.2.r.d 4
1620.2.r.e 4
1620.2.r.f 4
1620.2.r.g 8
1620.2.r.h 16
1620.2.u $$\chi_{1620}(181, \cdot)$$ 1620.2.u.a 30 6
1620.2.u.b 42
1620.2.x $$\chi_{1620}(53, \cdot)$$ 1620.2.x.a 8 4
1620.2.x.b 8
1620.2.x.c 16
1620.2.x.d 16
1620.2.x.e 24
1620.2.x.f 24
1620.2.y $$\chi_{1620}(703, \cdot)$$ n/a 560 4
1620.2.bb $$\chi_{1620}(179, \cdot)$$ n/a 624 6
1620.2.bd $$\chi_{1620}(289, \cdot)$$ n/a 108 6
1620.2.be $$\chi_{1620}(71, \cdot)$$ n/a 432 6
1620.2.bg $$\chi_{1620}(61, \cdot)$$ n/a 648 18
1620.2.bi $$\chi_{1620}(127, \cdot)$$ n/a 1248 12
1620.2.bk $$\chi_{1620}(17, \cdot)$$ n/a 216 12
1620.2.bm $$\chi_{1620}(59, \cdot)$$ n/a 5760 18
1620.2.bo $$\chi_{1620}(11, \cdot)$$ n/a 3888 18
1620.2.br $$\chi_{1620}(49, \cdot)$$ n/a 972 18
1620.2.bs $$\chi_{1620}(77, \cdot)$$ n/a 1944 36
1620.2.bv $$\chi_{1620}(7, \cdot)$$ n/a 11520 36

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1620))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1620)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(540))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(810))$$$$^{\oplus 2}$$