## Defining parameters

 Level: $$N$$ = $$1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Sturm bound: $$225792$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(1470))$$.

Total New Old
Modular forms 58368 12416 45952
Cusp forms 54529 12416 42113
Eisenstein series 3839 0 3839

## Trace form

 $$12416 q - 10 q^{3} - 16 q^{4} - 28 q^{5} - 26 q^{6} - 32 q^{7} - 32 q^{9} + O(q^{10})$$ $$12416 q - 10 q^{3} - 16 q^{4} - 28 q^{5} - 26 q^{6} - 32 q^{7} - 32 q^{9} - 20 q^{10} - 80 q^{11} - 2 q^{12} - 48 q^{13} + 10 q^{15} - 24 q^{17} + 56 q^{18} - 48 q^{19} + 4 q^{20} + 20 q^{21} + 8 q^{22} - 24 q^{23} + 22 q^{24} + 40 q^{25} + 48 q^{26} + 134 q^{27} + 24 q^{28} + 120 q^{29} + 106 q^{30} + 48 q^{31} + 152 q^{33} + 112 q^{34} + 96 q^{35} + 64 q^{36} + 352 q^{37} + 240 q^{38} + 352 q^{39} + 72 q^{40} + 376 q^{41} + 180 q^{42} + 272 q^{43} + 176 q^{44} + 200 q^{45} + 456 q^{46} + 336 q^{47} + 26 q^{48} + 576 q^{49} + 80 q^{50} + 396 q^{51} + 104 q^{52} + 240 q^{53} + 70 q^{54} + 460 q^{55} + 144 q^{56} + 152 q^{57} + 400 q^{58} + 376 q^{59} + 60 q^{60} + 552 q^{61} + 96 q^{62} + 84 q^{63} - 16 q^{64} + 120 q^{65} + 24 q^{66} + 336 q^{67} - 24 q^{68} + 112 q^{69} - 56 q^{72} + 232 q^{73} - 104 q^{74} + 158 q^{75} - 32 q^{76} - 72 q^{77} - 84 q^{78} - 8 q^{79} - 28 q^{80} - 24 q^{81} - 256 q^{82} + 144 q^{83} - 52 q^{84} - 184 q^{86} + 44 q^{87} - 40 q^{88} + 80 q^{89} - 204 q^{90} + 112 q^{91} - 72 q^{92} + 104 q^{93} - 256 q^{94} + 144 q^{95} - 18 q^{96} + 48 q^{97} - 48 q^{98} - 40 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(1470))$$

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
1470.2.a $$\chi_{1470}(1, \cdot)$$ 1470.2.a.a 1 1
1470.2.a.b 1
1470.2.a.c 1
1470.2.a.d 1
1470.2.a.e 1
1470.2.a.f 1
1470.2.a.g 1
1470.2.a.h 1
1470.2.a.i 1
1470.2.a.j 1
1470.2.a.k 1
1470.2.a.l 1
1470.2.a.m 1
1470.2.a.n 1
1470.2.a.o 1
1470.2.a.p 1
1470.2.a.q 1
1470.2.a.r 1
1470.2.a.s 2
1470.2.a.t 2
1470.2.a.u 2
1470.2.a.v 2
1470.2.b $$\chi_{1470}(881, \cdot)$$ 1470.2.b.a 12 1
1470.2.b.b 12
1470.2.b.c 16
1470.2.b.d 16
1470.2.d $$\chi_{1470}(1469, \cdot)$$ 1470.2.d.a 4 1
1470.2.d.b 4
1470.2.d.c 4
1470.2.d.d 4
1470.2.d.e 8
1470.2.d.f 8
1470.2.d.g 24
1470.2.d.h 24
1470.2.g $$\chi_{1470}(589, \cdot)$$ 1470.2.g.a 2 1
1470.2.g.b 2
1470.2.g.c 2
1470.2.g.d 2
1470.2.g.e 2
1470.2.g.f 2
1470.2.g.g 2
1470.2.g.h 6
1470.2.g.i 6
1470.2.g.j 8
1470.2.g.k 8
1470.2.i $$\chi_{1470}(361, \cdot)$$ 1470.2.i.a 2 2
1470.2.i.b 2
1470.2.i.c 2
1470.2.i.d 2
1470.2.i.e 2
1470.2.i.f 2
1470.2.i.g 2
1470.2.i.h 2
1470.2.i.i 2
1470.2.i.j 2
1470.2.i.k 2
1470.2.i.l 2
1470.2.i.m 2
1470.2.i.n 2
1470.2.i.o 2
1470.2.i.p 2
1470.2.i.q 2
1470.2.i.r 2
1470.2.i.s 2
1470.2.i.t 2
1470.2.i.u 4
1470.2.i.v 4
1470.2.i.w 4
1470.2.i.x 4
1470.2.j $$\chi_{1470}(197, \cdot)$$ n/a 164 2
1470.2.m $$\chi_{1470}(97, \cdot)$$ 1470.2.m.a 8 2
1470.2.m.b 8
1470.2.m.c 16
1470.2.m.d 16
1470.2.m.e 16
1470.2.m.f 16
1470.2.n $$\chi_{1470}(79, \cdot)$$ 1470.2.n.a 4 2
1470.2.n.b 4
1470.2.n.c 4
1470.2.n.d 4
1470.2.n.e 4
1470.2.n.f 4
1470.2.n.g 4
1470.2.n.h 4
1470.2.n.i 4
1470.2.n.j 12
1470.2.n.k 16
1470.2.n.l 16
1470.2.r $$\chi_{1470}(521, \cdot)$$ n/a 104 2
1470.2.t $$\chi_{1470}(509, \cdot)$$ n/a 160 2
1470.2.u $$\chi_{1470}(211, \cdot)$$ n/a 240 6
1470.2.v $$\chi_{1470}(313, \cdot)$$ n/a 160 4
1470.2.y $$\chi_{1470}(263, \cdot)$$ n/a 320 4
1470.2.bb $$\chi_{1470}(169, \cdot)$$ n/a 336 6
1470.2.bc $$\chi_{1470}(209, \cdot)$$ n/a 672 6
1470.2.be $$\chi_{1470}(41, \cdot)$$ n/a 432 6
1470.2.bg $$\chi_{1470}(121, \cdot)$$ n/a 432 12
1470.2.bi $$\chi_{1470}(13, \cdot)$$ n/a 672 12
1470.2.bj $$\chi_{1470}(113, \cdot)$$ n/a 1344 12
1470.2.bm $$\chi_{1470}(59, \cdot)$$ n/a 1344 12
1470.2.bo $$\chi_{1470}(101, \cdot)$$ n/a 912 12
1470.2.bq $$\chi_{1470}(109, \cdot)$$ n/a 672 12
1470.2.bt $$\chi_{1470}(23, \cdot)$$ n/a 2688 24
1470.2.bu $$\chi_{1470}(73, \cdot)$$ n/a 1344 24

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(1470))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(1470)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(105))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(210))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(245))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(294))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(490))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(735))$$$$^{\oplus 2}$$