# Properties

 Label 1620.3 Level 1620 Weight 3 Dimension 58800 Nonzero newspaces 24 Sturm bound 419904 Trace bound 16

## Defining parameters

 Level: $$N$$ = $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$24$$ Sturm bound: $$419904$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1620))$$.

Total New Old
Modular forms 142128 59472 82656
Cusp forms 137808 58800 79008
Eisenstein series 4320 672 3648

## Trace form

 $$58800 q - 24 q^{2} - 40 q^{4} - 81 q^{5} - 108 q^{6} - 6 q^{7} - 18 q^{8} - 72 q^{9} + O(q^{10})$$ $$58800 q - 24 q^{2} - 40 q^{4} - 81 q^{5} - 108 q^{6} - 6 q^{7} - 18 q^{8} - 72 q^{9} - 79 q^{10} + 72 q^{11} - 36 q^{12} - 2 q^{13} + 66 q^{14} - 96 q^{16} - 36 q^{17} - 36 q^{18} + 144 q^{19} - 99 q^{20} + 54 q^{21} - 176 q^{22} + 126 q^{23} - 36 q^{24} - 147 q^{25} - 426 q^{26} - 54 q^{27} - 290 q^{28} - 102 q^{29} - 54 q^{30} - 162 q^{31} - 204 q^{32} - 450 q^{33} - 132 q^{34} - 162 q^{35} - 108 q^{36} - 368 q^{37} + 132 q^{38} - 145 q^{40} - 1188 q^{41} - 1026 q^{42} - 492 q^{43} - 1758 q^{44} - 540 q^{45} - 934 q^{46} - 1314 q^{47} - 1026 q^{48} - 810 q^{49} - 621 q^{50} - 252 q^{51} - 102 q^{52} - 648 q^{53} + 216 q^{54} - 138 q^{55} + 894 q^{56} + 360 q^{57} + 262 q^{58} + 1080 q^{59} + 765 q^{60} + 294 q^{61} + 2358 q^{62} + 1080 q^{63} + 1274 q^{64} + 639 q^{65} + 1764 q^{66} + 1224 q^{67} + 1896 q^{68} - 1080 q^{69} + 125 q^{70} - 648 q^{71} - 36 q^{72} - 62 q^{73} - 378 q^{74} - 4 q^{76} - 786 q^{77} + 126 q^{78} - 174 q^{79} - 834 q^{80} - 72 q^{81} - 204 q^{82} - 342 q^{83} + 126 q^{84} - 968 q^{85} + 504 q^{86} + 2016 q^{87} + 1192 q^{88} - 762 q^{89} + 1017 q^{90} - 774 q^{91} + 4818 q^{92} - 1728 q^{93} + 1942 q^{94} + 108 q^{95} + 2466 q^{96} - 1904 q^{97} + 6138 q^{98} + 504 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1620))$$

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
1620.3.b $$\chi_{1620}(809, \cdot)$$ 1620.3.b.a 24 1
1620.3.b.b 24
1620.3.c $$\chi_{1620}(811, \cdot)$$ n/a 192 1
1620.3.f $$\chi_{1620}(1459, \cdot)$$ n/a 280 1
1620.3.g $$\chi_{1620}(161, \cdot)$$ 1620.3.g.a 4 1
1620.3.g.b 12
1620.3.g.c 16
1620.3.l $$\chi_{1620}(973, \cdot)$$ 1620.3.l.a 2 2
1620.3.l.b 2
1620.3.l.c 10
1620.3.l.d 10
1620.3.l.e 24
1620.3.l.f 24
1620.3.l.g 24
1620.3.m $$\chi_{1620}(323, \cdot)$$ n/a 560 2
1620.3.o $$\chi_{1620}(701, \cdot)$$ 1620.3.o.a 4 2
1620.3.o.b 4
1620.3.o.c 4
1620.3.o.d 4
1620.3.o.e 8
1620.3.o.f 8
1620.3.o.g 32
1620.3.p $$\chi_{1620}(379, \cdot)$$ n/a 568 2
1620.3.s $$\chi_{1620}(271, \cdot)$$ n/a 384 2
1620.3.t $$\chi_{1620}(269, \cdot)$$ 1620.3.t.a 8 2
1620.3.t.b 8
1620.3.t.c 8
1620.3.t.d 8
1620.3.t.e 16
1620.3.t.f 48
1620.3.v $$\chi_{1620}(217, \cdot)$$ n/a 192 4
1620.3.w $$\chi_{1620}(107, \cdot)$$ n/a 1136 4
1620.3.z $$\chi_{1620}(89, \cdot)$$ n/a 216 6
1620.3.ba $$\chi_{1620}(91, \cdot)$$ n/a 864 6
1620.3.bc $$\chi_{1620}(341, \cdot)$$ n/a 144 6
1620.3.bf $$\chi_{1620}(19, \cdot)$$ n/a 1272 6
1620.3.bh $$\chi_{1620}(143, \cdot)$$ n/a 2544 12
1620.3.bj $$\chi_{1620}(37, \cdot)$$ n/a 432 12
1620.3.bl $$\chi_{1620}(29, \cdot)$$ n/a 1944 18
1620.3.bn $$\chi_{1620}(31, \cdot)$$ n/a 7776 18
1620.3.bp $$\chi_{1620}(79, \cdot)$$ n/a 11592 18
1620.3.bq $$\chi_{1620}(41, \cdot)$$ n/a 1296 18
1620.3.bt $$\chi_{1620}(23, \cdot)$$ n/a 23184 36
1620.3.bu $$\chi_{1620}(13, \cdot)$$ n/a 3888 36

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

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1620))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(1620)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 18}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(540))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(810))$$$$^{\oplus 2}$$