## Defining parameters

 Level: $$N$$ = $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$70$$ Sturm bound: $$34560$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(570))$$.

Total New Old
Modular forms 9216 1965 7251
Cusp forms 8065 1965 6100
Eisenstein series 1151 0 1151

## Trace form

 $$1965q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 5q^{6} + 8q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$1965q + q^{2} + 5q^{3} + q^{4} + 9q^{5} + 5q^{6} + 8q^{7} + q^{8} + q^{9} - 7q^{10} - 20q^{11} + q^{12} + 78q^{13} + 48q^{14} + 17q^{15} + q^{16} + 42q^{17} - 15q^{18} + 153q^{19} + 29q^{20} + 76q^{21} + 104q^{22} + 48q^{23} + 5q^{24} + 89q^{25} + 86q^{26} + 35q^{27} + 32q^{28} + 54q^{29} + 29q^{30} + 72q^{31} + q^{32} + 104q^{33} - 14q^{34} + 48q^{35} + q^{36} - 18q^{37} - 23q^{38} + 66q^{39} - 23q^{40} + 34q^{41} - 24q^{42} + 60q^{43} - 4q^{44} + 9q^{45} - 72q^{46} + 24q^{47} - 29q^{48} - 15q^{49} - 15q^{50} - 168q^{51} - 18q^{52} - 42q^{53} - 103q^{54} - 20q^{55} - 8q^{56} - 179q^{57} - 2q^{58} - 20q^{59} - 23q^{60} + 22q^{61} - 16q^{62} - 124q^{63} + q^{64} + 50q^{65} - 180q^{66} + 116q^{67} - 30q^{68} - 116q^{69} + 8q^{70} - 48q^{71} - q^{72} + 70q^{73} - 42q^{74} - 31q^{75} - 31q^{76} - 96q^{77} - 82q^{78} - 72q^{79} + 9q^{80} - 207q^{81} - 166q^{82} - 252q^{83} - 116q^{84} - 462q^{85} - 212q^{86} - 274q^{87} - 4q^{88} - 310q^{89} - 293q^{90} - 336q^{91} - 168q^{92} - 340q^{93} - 304q^{94} - 347q^{95} - 11q^{96} - 342q^{97} - 327q^{98} - 418q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(570))$$

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
570.2.a $$\chi_{570}(1, \cdot)$$ 570.2.a.a 1 1
570.2.a.b 1
570.2.a.c 1
570.2.a.d 1
570.2.a.e 1
570.2.a.f 1
570.2.a.g 1
570.2.a.h 1
570.2.a.i 1
570.2.a.j 1
570.2.a.k 1
570.2.a.l 1
570.2.a.m 1
570.2.c $$\chi_{570}(569, \cdot)$$ 570.2.c.a 4 1
570.2.c.b 4
570.2.c.c 8
570.2.c.d 8
570.2.c.e 8
570.2.c.f 8
570.2.d $$\chi_{570}(229, \cdot)$$ 570.2.d.a 2 1
570.2.d.b 2
570.2.d.c 6
570.2.d.d 6
570.2.f $$\chi_{570}(341, \cdot)$$ 570.2.f.a 4 1
570.2.f.b 4
570.2.f.c 8
570.2.f.d 8
570.2.i $$\chi_{570}(121, \cdot)$$ 570.2.i.a 2 2
570.2.i.b 2
570.2.i.c 2
570.2.i.d 2
570.2.i.e 2
570.2.i.f 4
570.2.i.g 4
570.2.i.h 4
570.2.i.i 4
570.2.i.j 6
570.2.k $$\chi_{570}(77, \cdot)$$ 570.2.k.a 36 2
570.2.k.b 36
570.2.m $$\chi_{570}(37, \cdot)$$ 570.2.m.a 20 2
570.2.m.b 20
570.2.n $$\chi_{570}(179, \cdot)$$ 570.2.n.a 80 2
570.2.q $$\chi_{570}(49, \cdot)$$ 570.2.q.a 8 2
570.2.q.b 12
570.2.q.c 20
570.2.s $$\chi_{570}(221, \cdot)$$ 570.2.s.a 24 2
570.2.s.b 24
570.2.u $$\chi_{570}(61, \cdot)$$ 570.2.u.a 6 6
570.2.u.b 6
570.2.u.c 6
570.2.u.d 6
570.2.u.e 6
570.2.u.f 6
570.2.u.g 6
570.2.u.h 6
570.2.u.i 12
570.2.u.j 12
570.2.v $$\chi_{570}(83, \cdot)$$ 570.2.v.a 8 4
570.2.v.b 8
570.2.v.c 144
570.2.x $$\chi_{570}(103, \cdot)$$ 570.2.x.a 40 4
570.2.x.b 40
570.2.bb $$\chi_{570}(41, \cdot)$$ 570.2.bb.a 84 6
570.2.bb.b 84
570.2.bc $$\chi_{570}(139, \cdot)$$ 570.2.bc.a 60 6
570.2.bc.b 60
570.2.bf $$\chi_{570}(29, \cdot)$$ 570.2.bf.a 240 6
570.2.bh $$\chi_{570}(13, \cdot)$$ 570.2.bh.a 120 12
570.2.bh.b 120
570.2.bi $$\chi_{570}(17, \cdot)$$ 570.2.bi.a 480 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(570))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(570)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(95))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(190))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(285))$$$$^{\oplus 2}$$