## Defining parameters

 Level: $$N$$ = $$1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$36$$ Sturm bound: $$248832$$ Trace bound: $$27$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1530))$$.

Total New Old
Modular forms 64256 14942 49314
Cusp forms 60161 14942 45219
Eisenstein series 4095 0 4095

## Trace form

 $$14942 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 10 q^{5} + 12 q^{6} - 24 q^{7} + 6 q^{8} + 28 q^{9} + O(q^{10})$$ $$14942 q - 6 q^{2} - 12 q^{3} - 6 q^{4} - 10 q^{5} + 12 q^{6} - 24 q^{7} + 6 q^{8} + 28 q^{9} + 18 q^{10} + 20 q^{11} + 16 q^{12} - 4 q^{13} + 8 q^{14} + 48 q^{15} - 14 q^{16} - 14 q^{17} + 8 q^{18} - 16 q^{19} + 18 q^{20} + 72 q^{21} - 12 q^{22} + 40 q^{23} - 12 q^{24} - 2 q^{25} - 20 q^{26} + 48 q^{27} - 28 q^{29} - 48 q^{31} - 6 q^{32} - 20 q^{33} - 4 q^{34} - 64 q^{35} - 20 q^{36} - 12 q^{37} - 36 q^{38} + 6 q^{40} + 96 q^{41} + 64 q^{42} + 124 q^{43} + 128 q^{44} - 32 q^{45} + 192 q^{46} + 360 q^{47} - 20 q^{48} + 378 q^{49} - 46 q^{50} + 226 q^{51} + 172 q^{52} + 316 q^{53} + 164 q^{54} + 216 q^{55} - 40 q^{56} + 268 q^{57} + 124 q^{58} + 332 q^{59} + 80 q^{60} + 132 q^{61} + 288 q^{62} + 272 q^{63} + 6 q^{64} + 176 q^{65} + 128 q^{66} + 4 q^{67} + 44 q^{68} + 128 q^{69} + 144 q^{70} + 176 q^{71} + 12 q^{72} + 268 q^{73} + 172 q^{74} - 12 q^{75} + 92 q^{76} + 120 q^{77} + 56 q^{78} + 160 q^{79} + 38 q^{80} - 4 q^{81} + 252 q^{82} + 40 q^{83} + 24 q^{84} + 410 q^{85} + 204 q^{86} + 232 q^{87} + 84 q^{88} + 476 q^{89} + 112 q^{90} + 704 q^{91} + 104 q^{92} + 192 q^{93} + 184 q^{94} + 408 q^{95} + 16 q^{96} + 408 q^{97} + 126 q^{98} + 328 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1530))$$

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
1530.2.a $$\chi_{1530}(1, \cdot)$$ 1530.2.a.a 1 1
1530.2.a.b 1
1530.2.a.c 1
1530.2.a.d 1
1530.2.a.e 1
1530.2.a.f 1
1530.2.a.g 1
1530.2.a.h 1
1530.2.a.i 1
1530.2.a.j 1
1530.2.a.k 1
1530.2.a.l 1
1530.2.a.m 1
1530.2.a.n 1
1530.2.a.o 1
1530.2.a.p 1
1530.2.a.q 2
1530.2.a.r 2
1530.2.a.s 2
1530.2.a.t 2
1530.2.c $$\chi_{1530}(271, \cdot)$$ 1530.2.c.a 2 1
1530.2.c.b 2
1530.2.c.c 2
1530.2.c.d 2
1530.2.c.e 2
1530.2.c.f 4
1530.2.c.g 4
1530.2.c.h 6
1530.2.c.i 6
1530.2.d $$\chi_{1530}(919, \cdot)$$ 1530.2.d.a 2 1
1530.2.d.b 2
1530.2.d.c 2
1530.2.d.d 2
1530.2.d.e 4
1530.2.d.f 4
1530.2.d.g 6
1530.2.d.h 6
1530.2.d.i 6
1530.2.d.j 6
1530.2.f $$\chi_{1530}(1189, \cdot)$$ 1530.2.f.a 2 1
1530.2.f.b 2
1530.2.f.c 2
1530.2.f.d 2
1530.2.f.e 2
1530.2.f.f 2
1530.2.f.g 4
1530.2.f.h 4
1530.2.f.i 4
1530.2.f.j 6
1530.2.f.k 6
1530.2.f.l 8
1530.2.i $$\chi_{1530}(511, \cdot)$$ n/a 128 2
1530.2.j $$\chi_{1530}(863, \cdot)$$ 1530.2.j.a 4 2
1530.2.j.b 28
1530.2.j.c 40
1530.2.m $$\chi_{1530}(647, \cdot)$$ 1530.2.m.a 4 2
1530.2.m.b 4
1530.2.m.c 4
1530.2.m.d 4
1530.2.m.e 4
1530.2.m.f 4
1530.2.m.g 8
1530.2.m.h 16
1530.2.m.i 16
1530.2.n $$\chi_{1530}(829, \cdot)$$ 1530.2.n.a 2 2
1530.2.n.b 2
1530.2.n.c 2
1530.2.n.d 2
1530.2.n.e 2
1530.2.n.f 2
1530.2.n.g 2
1530.2.n.h 2
1530.2.n.i 4
1530.2.n.j 4
1530.2.n.k 4
1530.2.n.l 4
1530.2.n.m 4
1530.2.n.n 4
1530.2.n.o 4
1530.2.n.p 4
1530.2.n.q 8
1530.2.n.r 8
1530.2.n.s 12
1530.2.n.t 12
1530.2.q $$\chi_{1530}(361, \cdot)$$ 1530.2.q.a 4 2
1530.2.q.b 4
1530.2.q.c 4
1530.2.q.d 4
1530.2.q.e 4
1530.2.q.f 8
1530.2.q.g 8
1530.2.q.h 8
1530.2.q.i 8
1530.2.q.j 8
1530.2.r $$\chi_{1530}(917, \cdot)$$ 1530.2.r.a 4 2
1530.2.r.b 4
1530.2.r.c 24
1530.2.r.d 40
1530.2.u $$\chi_{1530}(557, \cdot)$$ 1530.2.u.a 4 2
1530.2.u.b 28
1530.2.u.c 40
1530.2.x $$\chi_{1530}(169, \cdot)$$ n/a 216 2
1530.2.z $$\chi_{1530}(409, \cdot)$$ n/a 192 2
1530.2.ba $$\chi_{1530}(781, \cdot)$$ n/a 144 2
1530.2.bc $$\chi_{1530}(451, \cdot)$$ n/a 120 4
1530.2.be $$\chi_{1530}(287, \cdot)$$ n/a 144 4
1530.2.bh $$\chi_{1530}(53, \cdot)$$ n/a 144 4
1530.2.bj $$\chi_{1530}(19, \cdot)$$ n/a 184 4
1530.2.bk $$\chi_{1530}(47, \cdot)$$ n/a 432 4
1530.2.bn $$\chi_{1530}(203, \cdot)$$ n/a 432 4
1530.2.bo $$\chi_{1530}(421, \cdot)$$ n/a 288 4
1530.2.br $$\chi_{1530}(259, \cdot)$$ n/a 432 4
1530.2.bs $$\chi_{1530}(137, \cdot)$$ n/a 384 4
1530.2.bv $$\chi_{1530}(353, \cdot)$$ n/a 432 4
1530.2.bx $$\chi_{1530}(73, \cdot)$$ n/a 360 8
1530.2.bz $$\chi_{1530}(71, \cdot)$$ n/a 192 8
1530.2.cb $$\chi_{1530}(269, \cdot)$$ n/a 288 8
1530.2.cc $$\chi_{1530}(37, \cdot)$$ n/a 360 8
1530.2.ce $$\chi_{1530}(49, \cdot)$$ n/a 864 8
1530.2.ch $$\chi_{1530}(77, \cdot)$$ n/a 864 8
1530.2.ci $$\chi_{1530}(257, \cdot)$$ n/a 864 8
1530.2.cl $$\chi_{1530}(121, \cdot)$$ n/a 576 8
1530.2.cn $$\chi_{1530}(97, \cdot)$$ n/a 1728 16
1530.2.co $$\chi_{1530}(29, \cdot)$$ n/a 1728 16
1530.2.cq $$\chi_{1530}(11, \cdot)$$ n/a 1152 16
1530.2.cs $$\chi_{1530}(7, \cdot)$$ n/a 1728 16

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1530))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1530)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(85))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(90))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(153))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(170))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(255))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(306))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(510))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(765))$$$$^{\oplus 2}$$