## Defining parameters

 Level: $$N$$ = $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$24$$ Newform subspaces: $$88$$ Sturm bound: $$69120$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(770))$$.

Total New Old
Modular forms 18240 4969 13271
Cusp forms 16321 4969 11352
Eisenstein series 1919 0 1919

## Trace form

 $$4969q - 3q^{2} - 4q^{3} + 5q^{4} + 9q^{5} + 32q^{6} + 37q^{7} - 3q^{8} + 97q^{9} + O(q^{10})$$ $$4969q - 3q^{2} - 4q^{3} + 5q^{4} + 9q^{5} + 32q^{6} + 37q^{7} - 3q^{8} + 97q^{9} + 29q^{10} + 41q^{11} + 36q^{12} + 38q^{13} + 29q^{14} + 96q^{15} + 5q^{16} + 122q^{17} + 29q^{18} + 88q^{19} + 9q^{20} + 84q^{21} + q^{22} + 104q^{23} + 32q^{24} + 61q^{25} + 62q^{26} + 92q^{27} - 11q^{28} + 70q^{29} - 48q^{30} + 40q^{31} + 17q^{32} + 104q^{33} - 22q^{34} - 15q^{35} - 59q^{36} - 18q^{37} + 36q^{38} + 48q^{39} + 29q^{40} + 130q^{41} - 64q^{42} + 20q^{43} - 43q^{44} - 167q^{45} - 104q^{46} - 128q^{47} - 4q^{48} + 21q^{49} - 51q^{50} - 36q^{51} - 82q^{52} + 38q^{53} - 136q^{54} - 155q^{55} + 9q^{56} - 12q^{57} - 106q^{58} - 40q^{59} - 84q^{60} - 210q^{61} - 112q^{62} - 339q^{63} + 5q^{64} - 238q^{65} - 396q^{66} - 260q^{67} - 78q^{68} - 416q^{69} - 111q^{70} - 184q^{71} - 151q^{72} - 230q^{73} - 130q^{74} - 256q^{75} - 52q^{76} - 199q^{77} - 296q^{78} - 240q^{79} - 31q^{80} - 487q^{81} - 82q^{82} - 136q^{83} - 116q^{84} + 46q^{85} - 136q^{86} - 224q^{87} - 103q^{88} - 94q^{89} - 135q^{90} - 226q^{91} + 48q^{92} - 256q^{93} + 24q^{94} - 204q^{95} - 36q^{96} - 90q^{97} - 175q^{98} - 59q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(770))$$

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
770.2.a $$\chi_{770}(1, \cdot)$$ 770.2.a.a 1 1
770.2.a.b 1
770.2.a.c 1
770.2.a.d 1
770.2.a.e 1
770.2.a.f 1
770.2.a.g 1
770.2.a.h 2
770.2.a.i 2
770.2.a.j 2
770.2.a.k 2
770.2.a.l 3
770.2.a.m 3
770.2.c $$\chi_{770}(309, \cdot)$$ 770.2.c.a 2 1
770.2.c.b 2
770.2.c.c 2
770.2.c.d 4
770.2.c.e 8
770.2.c.f 10
770.2.e $$\chi_{770}(461, \cdot)$$ 770.2.e.a 16 1
770.2.e.b 16
770.2.g $$\chi_{770}(769, \cdot)$$ 770.2.g.a 24 1
770.2.g.b 24
770.2.i $$\chi_{770}(221, \cdot)$$ 770.2.i.a 2 2
770.2.i.b 2
770.2.i.c 2
770.2.i.d 2
770.2.i.e 2
770.2.i.f 2
770.2.i.g 2
770.2.i.h 2
770.2.i.i 2
770.2.i.j 6
770.2.i.k 6
770.2.i.l 8
770.2.i.m 10
770.2.l $$\chi_{770}(573, \cdot)$$ 770.2.l.a 16 2
770.2.l.b 24
770.2.l.c 40
770.2.m $$\chi_{770}(43, \cdot)$$ 770.2.m.a 4 2
770.2.m.b 4
770.2.m.c 4
770.2.m.d 8
770.2.m.e 16
770.2.m.f 36
770.2.n $$\chi_{770}(71, \cdot)$$ 770.2.n.a 4 4
770.2.n.b 4
770.2.n.c 4
770.2.n.d 4
770.2.n.e 8
770.2.n.f 8
770.2.n.g 12
770.2.n.h 12
770.2.n.i 12
770.2.n.j 12
770.2.n.k 16
770.2.o $$\chi_{770}(439, \cdot)$$ 770.2.o.a 48 2
770.2.o.b 48
770.2.r $$\chi_{770}(529, \cdot)$$ 770.2.r.a 40 2
770.2.r.b 40
770.2.t $$\chi_{770}(131, \cdot)$$ 770.2.t.a 32 2
770.2.t.b 32
770.2.w $$\chi_{770}(139, \cdot)$$ 770.2.w.a 96 4
770.2.w.b 96
770.2.y $$\chi_{770}(41, \cdot)$$ 770.2.y.a 64 4
770.2.y.b 64
770.2.ba $$\chi_{770}(169, \cdot)$$ 770.2.ba.a 8 4
770.2.ba.b 64
770.2.ba.c 72
770.2.bc $$\chi_{770}(243, \cdot)$$ 770.2.bc.a 80 4
770.2.bc.b 80
770.2.bd $$\chi_{770}(263, \cdot)$$ 770.2.bd.a 192 4
770.2.bg $$\chi_{770}(81, \cdot)$$ 770.2.bg.a 8 8
770.2.bg.b 8
770.2.bg.c 48
770.2.bg.d 48
770.2.bg.e 72
770.2.bg.f 72
770.2.bh $$\chi_{770}(57, \cdot)$$ 770.2.bh.a 144 8
770.2.bh.b 144
770.2.bi $$\chi_{770}(27, \cdot)$$ 770.2.bi.a 384 8
770.2.bm $$\chi_{770}(61, \cdot)$$ 770.2.bm.a 128 8
770.2.bm.b 128
770.2.bo $$\chi_{770}(9, \cdot)$$ 770.2.bo.a 384 8
770.2.br $$\chi_{770}(19, \cdot)$$ 770.2.br.a 192 8
770.2.br.b 192
770.2.bu $$\chi_{770}(107, \cdot)$$ 770.2.bu.a 768 16
770.2.bv $$\chi_{770}(3, \cdot)$$ 770.2.bv.a 768 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(770))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(770)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(55))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(110))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(154))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(385))$$$$^{\oplus 2}$$