# Properties

 Label 370.2 Level 370 Weight 2 Dimension 1255 Nonzero newspaces 18 Newform subspaces 70 Sturm bound 16416 Trace bound 7

## Defining parameters

 Level: $$N$$ = $$370 = 2 \cdot 5 \cdot 37$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$70$$ Sturm bound: $$16416$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(370))$$.

Total New Old
Modular forms 4392 1255 3137
Cusp forms 3817 1255 2562
Eisenstein series 575 0 575

## Trace form

 $$1255 q + q^{2} + 4 q^{3} + q^{4} + q^{5} + 4 q^{6} + 8 q^{7} + q^{8} + 13 q^{9} + O(q^{10})$$ $$1255 q + q^{2} + 4 q^{3} + q^{4} + q^{5} + 4 q^{6} + 8 q^{7} + q^{8} + 13 q^{9} + q^{10} + 12 q^{11} + 4 q^{12} + 14 q^{13} + 8 q^{14} + 4 q^{15} + q^{16} + 18 q^{17} + 13 q^{18} + 20 q^{19} + q^{20} + 32 q^{21} + 12 q^{22} + 24 q^{23} + 4 q^{24} + q^{25} - 4 q^{26} - 56 q^{27} - 16 q^{28} - 42 q^{29} - 68 q^{30} - 184 q^{31} + q^{32} - 96 q^{33} - 126 q^{34} - 100 q^{35} - 77 q^{36} - 131 q^{37} - 52 q^{38} - 112 q^{39} - 44 q^{40} - 174 q^{41} - 112 q^{42} - 100 q^{43} + 12 q^{44} - 95 q^{45} - 120 q^{46} - 24 q^{47} - 20 q^{48} - 39 q^{49} - 8 q^{50} + 72 q^{51} + 14 q^{52} + 54 q^{53} + 40 q^{54} + 12 q^{55} + 8 q^{56} + 80 q^{57} + 30 q^{58} - 12 q^{59} + 4 q^{60} - 28 q^{61} + 32 q^{62} - 112 q^{63} + q^{64} - 67 q^{65} + 48 q^{66} - 4 q^{67} + 18 q^{68} - 192 q^{69} + 8 q^{70} - 72 q^{71} + 13 q^{72} - 70 q^{73} + 37 q^{74} - 176 q^{75} + 20 q^{76} - 48 q^{77} + 56 q^{78} - 64 q^{79} + q^{80} - 167 q^{81} + 42 q^{82} + 12 q^{83} + 32 q^{84} - 63 q^{85} + 44 q^{86} - 96 q^{87} + 12 q^{88} + 13 q^{90} + 16 q^{91} - 48 q^{92} - 88 q^{93} - 96 q^{94} - 124 q^{95} + 4 q^{96} - 190 q^{97} - 231 q^{98} - 204 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(370))$$

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
370.2.a $$\chi_{370}(1, \cdot)$$ 370.2.a.a 1 1
370.2.a.b 1
370.2.a.c 1
370.2.a.d 1
370.2.a.e 2
370.2.a.f 2
370.2.a.g 3
370.2.b $$\chi_{370}(149, \cdot)$$ 370.2.b.a 2 1
370.2.b.b 2
370.2.b.c 4
370.2.b.d 10
370.2.c $$\chi_{370}(369, \cdot)$$ 370.2.c.a 10 1
370.2.c.b 10
370.2.d $$\chi_{370}(221, \cdot)$$ 370.2.d.a 2 1
370.2.d.b 2
370.2.d.c 6
370.2.e $$\chi_{370}(121, \cdot)$$ 370.2.e.a 2 2
370.2.e.b 2
370.2.e.c 2
370.2.e.d 4
370.2.e.e 4
370.2.e.f 6
370.2.g $$\chi_{370}(43, \cdot)$$ 370.2.g.a 2 2
370.2.g.b 2
370.2.g.c 4
370.2.g.d 10
370.2.g.e 20
370.2.h $$\chi_{370}(117, \cdot)$$ 370.2.h.a 2 2
370.2.h.b 2
370.2.h.c 4
370.2.h.d 10
370.2.h.e 20
370.2.l $$\chi_{370}(11, \cdot)$$ 370.2.l.a 4 2
370.2.l.b 4
370.2.l.c 12
370.2.m $$\chi_{370}(159, \cdot)$$ 370.2.m.a 4 2
370.2.m.b 4
370.2.m.c 16
370.2.m.d 16
370.2.n $$\chi_{370}(269, \cdot)$$ 370.2.n.a 4 2
370.2.n.b 4
370.2.n.c 4
370.2.n.d 4
370.2.n.e 8
370.2.n.f 12
370.2.o $$\chi_{370}(71, \cdot)$$ 370.2.o.a 18 6
370.2.o.b 18
370.2.o.c 24
370.2.o.d 24
370.2.q $$\chi_{370}(97, \cdot)$$ 370.2.q.a 4 4
370.2.q.b 4
370.2.q.c 8
370.2.q.d 12
370.2.q.e 16
370.2.q.f 32
370.2.r $$\chi_{370}(23, \cdot)$$ 370.2.r.a 4 4
370.2.r.b 4
370.2.r.c 8
370.2.r.d 12
370.2.r.e 16
370.2.r.f 32
370.2.v $$\chi_{370}(99, \cdot)$$ 370.2.v.a 60 6
370.2.v.b 60
370.2.w $$\chi_{370}(21, \cdot)$$ 370.2.w.a 36 6
370.2.w.b 48
370.2.x $$\chi_{370}(9, \cdot)$$ 370.2.x.a 108 6
370.2.ba $$\chi_{370}(17, \cdot)$$ 370.2.ba.a 108 12
370.2.ba.b 120
370.2.bd $$\chi_{370}(13, \cdot)$$ 370.2.bd.a 108 12
370.2.bd.b 120

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(370))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(370)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(37))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(74))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(185))$$$$^{\oplus 2}$$