## Defining parameters

 Level: $$N$$ = $$1620 = 2^{2} \cdot 3^{4} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$559872$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1620))$$.

Total New Old
Modular forms 212112 89040 123072
Cusp forms 207792 88368 119424
Eisenstein series 4320 672 3648

## Trace form

 $$88368 q - 24 q^{2} - 40 q^{4} - 60 q^{5} - 108 q^{6} - 36 q^{7} - 30 q^{8} - 72 q^{9} + O(q^{10})$$ $$88368 q - 24 q^{2} - 40 q^{4} - 60 q^{5} - 108 q^{6} - 36 q^{7} - 30 q^{8} - 72 q^{9} - 103 q^{10} - 174 q^{11} - 36 q^{12} - 44 q^{13} + 126 q^{14} + 240 q^{16} + 348 q^{17} - 36 q^{18} - 540 q^{19} + 195 q^{20} - 432 q^{21} - 80 q^{22} + 948 q^{23} - 36 q^{24} + 150 q^{25} - 42 q^{26} + 1404 q^{27} + 190 q^{28} + 1500 q^{29} - 54 q^{30} + 432 q^{31} - 1404 q^{32} - 936 q^{33} - 1116 q^{34} - 2892 q^{35} - 108 q^{36} - 1088 q^{37} - 1812 q^{38} + 545 q^{40} + 7134 q^{41} + 7074 q^{42} + 3150 q^{43} + 10638 q^{44} + 918 q^{45} + 2090 q^{46} - 2004 q^{47} - 4590 q^{48} - 2928 q^{49} - 8031 q^{50} - 5922 q^{51} - 6210 q^{52} - 9528 q^{53} - 13392 q^{54} - 4932 q^{55} - 17754 q^{56} - 4500 q^{57} - 1502 q^{58} - 7326 q^{59} - 1341 q^{60} - 600 q^{61} + 7494 q^{62} + 3996 q^{63} + 7610 q^{64} + 9354 q^{65} + 14724 q^{66} + 378 q^{67} + 19308 q^{68} + 14148 q^{69} + 551 q^{70} + 1200 q^{71} - 36 q^{72} + 2260 q^{73} + 906 q^{74} + 452 q^{76} - 8208 q^{77} - 522 q^{78} + 8568 q^{79} + 132 q^{80} - 11736 q^{81} - 1116 q^{82} - 7368 q^{83} + 450 q^{84} + 2098 q^{85} - 48 q^{86} - 10944 q^{87} - 8648 q^{88} + 14916 q^{89} - 12591 q^{90} + 2916 q^{91} - 49386 q^{92} + 36828 q^{93} - 14426 q^{94} + 3276 q^{95} - 5958 q^{96} - 9674 q^{97} + 7722 q^{98} + 15084 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1620))$$

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
1620.4.a $$\chi_{1620}(1, \cdot)$$ 1620.4.a.a 1 1
1620.4.a.b 1
1620.4.a.c 3
1620.4.a.d 3
1620.4.a.e 3
1620.4.a.f 3
1620.4.a.g 4
1620.4.a.h 4
1620.4.a.i 6
1620.4.a.j 6
1620.4.a.k 7
1620.4.a.l 7
1620.4.d $$\chi_{1620}(649, \cdot)$$ 1620.4.d.a 10 1
1620.4.d.b 10
1620.4.d.c 16
1620.4.d.d 18
1620.4.d.e 18
1620.4.e $$\chi_{1620}(971, \cdot)$$ n/a 288 1
1620.4.h $$\chi_{1620}(1619, \cdot)$$ n/a 424 1
1620.4.i $$\chi_{1620}(541, \cdot)$$ 1620.4.i.a 2 2
1620.4.i.b 2
1620.4.i.c 2
1620.4.i.d 2
1620.4.i.e 2
1620.4.i.f 2
1620.4.i.g 2
1620.4.i.h 2
1620.4.i.i 2
1620.4.i.j 2
1620.4.i.k 2
1620.4.i.l 2
1620.4.i.m 4
1620.4.i.n 4
1620.4.i.o 4
1620.4.i.p 4
1620.4.i.q 4
1620.4.i.r 4
1620.4.i.s 6
1620.4.i.t 6
1620.4.i.u 6
1620.4.i.v 6
1620.4.i.w 12
1620.4.i.x 12
1620.4.j $$\chi_{1620}(1133, \cdot)$$ n/a 144 2
1620.4.k $$\chi_{1620}(163, \cdot)$$ n/a 848 2
1620.4.n $$\chi_{1620}(539, \cdot)$$ n/a 856 2
1620.4.q $$\chi_{1620}(431, \cdot)$$ n/a 576 2
1620.4.r $$\chi_{1620}(109, \cdot)$$ n/a 144 2
1620.4.u $$\chi_{1620}(181, \cdot)$$ n/a 216 6
1620.4.x $$\chi_{1620}(53, \cdot)$$ n/a 288 4
1620.4.y $$\chi_{1620}(703, \cdot)$$ n/a 1712 4
1620.4.bb $$\chi_{1620}(179, \cdot)$$ n/a 1920 6
1620.4.bd $$\chi_{1620}(289, \cdot)$$ n/a 324 6
1620.4.be $$\chi_{1620}(71, \cdot)$$ n/a 1296 6
1620.4.bg $$\chi_{1620}(61, \cdot)$$ n/a 1944 18
1620.4.bi $$\chi_{1620}(127, \cdot)$$ n/a 3840 12
1620.4.bk $$\chi_{1620}(17, \cdot)$$ n/a 648 12
1620.4.bm $$\chi_{1620}(59, \cdot)$$ n/a 17424 18
1620.4.bo $$\chi_{1620}(11, \cdot)$$ n/a 11664 18
1620.4.br $$\chi_{1620}(49, \cdot)$$ n/a 2916 18
1620.4.bs $$\chi_{1620}(77, \cdot)$$ n/a 5832 36
1620.4.bv $$\chi_{1620}(7, \cdot)$$ n/a 34848 36

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1620))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1620)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 15}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 16}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 18}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(60))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(162))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(270))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(324))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(405))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(540))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(810))$$$$^{\oplus 2}$$