# Properties

 Label 385.2 Level 385 Weight 2 Dimension 4711 Nonzero newspaces 24 Newform subspaces 54 Sturm bound 23040 Trace bound 5

## Defining parameters

 Level: $$N$$ = $$385 = 5 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$54$$ Sturm bound: $$23040$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(385))$$.

Total New Old
Modular forms 6240 5247 993
Cusp forms 5281 4711 570
Eisenstein series 959 536 423

## Trace form

 $$4711q - 19q^{2} - 20q^{3} - 23q^{4} - 45q^{5} - 104q^{6} - 49q^{7} - 107q^{8} - 69q^{9} + O(q^{10})$$ $$4711q - 19q^{2} - 20q^{3} - 23q^{4} - 45q^{5} - 104q^{6} - 49q^{7} - 107q^{8} - 69q^{9} - 93q^{10} - 117q^{11} - 164q^{12} - 50q^{13} - 95q^{14} - 178q^{15} - 223q^{16} - 94q^{17} - 175q^{18} - 96q^{19} - 169q^{20} - 204q^{21} - 199q^{22} - 140q^{23} - 200q^{24} - 121q^{25} - 246q^{26} - 104q^{27} - 131q^{28} - 134q^{29} - 130q^{30} - 136q^{31} - 135q^{32} - 132q^{33} - 150q^{34} - 93q^{35} - 463q^{36} - 102q^{37} - 200q^{38} - 172q^{39} - 37q^{40} - 246q^{41} - 188q^{42} - 148q^{43} - 107q^{44} - 161q^{45} - 196q^{46} - 60q^{47} - 136q^{48} - 133q^{49} - 199q^{50} - 244q^{51} - 38q^{52} - 170q^{53} - 132q^{54} - 33q^{55} - 411q^{56} - 176q^{57} + 66q^{58} - 60q^{59} + 122q^{60} + 50q^{61} - 36q^{62} + 87q^{63} + 149q^{64} + 42q^{65} + 176q^{66} + 48q^{67} + 190q^{68} + 124q^{69} + 187q^{70} - 220q^{71} + 397q^{72} + 22q^{73} - 86q^{74} + 20q^{75} + 196q^{76} - 29q^{77} - 152q^{78} - 64q^{79} - 73q^{80} - 29q^{81} - 82q^{82} - 164q^{83} + 316q^{84} - 258q^{85} - 160q^{86} - 176q^{87} - 155q^{88} - 166q^{89} + 99q^{90} - 198q^{91} - 152q^{92} - 76q^{93} - 188q^{94} - 64q^{95} + 32q^{96} - 130q^{97} - 59q^{98} - 153q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(385))$$

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
385.2.a $$\chi_{385}(1, \cdot)$$ 385.2.a.a 1 1
385.2.a.b 1
385.2.a.c 2
385.2.a.d 2
385.2.a.e 3
385.2.a.f 3
385.2.a.g 3
385.2.a.h 4
385.2.b $$\chi_{385}(309, \cdot)$$ 385.2.b.a 2 1
385.2.b.b 2
385.2.b.c 12
385.2.b.d 16
385.2.c $$\chi_{385}(76, \cdot)$$ 385.2.c.a 32 1
385.2.h $$\chi_{385}(384, \cdot)$$ 385.2.h.a 4 1
385.2.h.b 8
385.2.h.c 32
385.2.i $$\chi_{385}(221, \cdot)$$ 385.2.i.a 8 2
385.2.i.b 12
385.2.i.c 16
385.2.i.d 20
385.2.j $$\chi_{385}(188, \cdot)$$ 385.2.j.a 4 2
385.2.j.b 4
385.2.j.c 32
385.2.j.d 40
385.2.k $$\chi_{385}(43, \cdot)$$ 385.2.k.a 72 2
385.2.n $$\chi_{385}(36, \cdot)$$ 385.2.n.a 4 4
385.2.n.b 4
385.2.n.c 8
385.2.n.d 16
385.2.n.e 28
385.2.n.f 36
385.2.o $$\chi_{385}(54, \cdot)$$ 385.2.o.a 16 2
385.2.o.b 72
385.2.t $$\chi_{385}(144, \cdot)$$ 385.2.t.a 4 2
385.2.t.b 36
385.2.t.c 40
385.2.u $$\chi_{385}(131, \cdot)$$ 385.2.u.a 64 2
385.2.v $$\chi_{385}(139, \cdot)$$ 385.2.v.a 8 4
385.2.v.b 8
385.2.v.c 160
385.2.ba $$\chi_{385}(6, \cdot)$$ 385.2.ba.a 128 4
385.2.bb $$\chi_{385}(64, \cdot)$$ 385.2.bb.a 144 4
385.2.bc $$\chi_{385}(32, \cdot)$$ 385.2.bc.a 176 4
385.2.bd $$\chi_{385}(12, \cdot)$$ 385.2.bd.a 80 4
385.2.bd.b 80
385.2.bg $$\chi_{385}(16, \cdot)$$ 385.2.bg.a 128 8
385.2.bg.b 128
385.2.bj $$\chi_{385}(8, \cdot)$$ 385.2.bj.a 288 8
385.2.bk $$\chi_{385}(27, \cdot)$$ 385.2.bk.a 352 8
385.2.bl $$\chi_{385}(61, \cdot)$$ 385.2.bl.a 256 8
385.2.bm $$\chi_{385}(4, \cdot)$$ 385.2.bm.a 352 8
385.2.br $$\chi_{385}(19, \cdot)$$ 385.2.br.a 352 8
385.2.bu $$\chi_{385}(3, \cdot)$$ 385.2.bu.a 704 16
385.2.bv $$\chi_{385}(2, \cdot)$$ 385.2.bv.a 704 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(385))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(385)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 2}$$