# Properties

 Label 77.4 Level 77 Weight 4 Dimension 634 Nonzero newspaces 8 Newform subspaces 17 Sturm bound 1920 Trace bound 2

## Defining parameters

 Level: $$N$$ = $$77 = 7 \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Newform subspaces: $$17$$ Sturm bound: $$1920$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(77))$$.

Total New Old
Modular forms 780 726 54
Cusp forms 660 634 26
Eisenstein series 120 92 28

## Trace form

 $$634 q - 14 q^{2} - 2 q^{3} - 14 q^{4} - 38 q^{5} + 20 q^{6} - 57 q^{7} - 64 q^{8} - 90 q^{9} + O(q^{10})$$ $$634 q - 14 q^{2} - 2 q^{3} - 14 q^{4} - 38 q^{5} + 20 q^{6} - 57 q^{7} - 64 q^{8} - 90 q^{9} - 150 q^{10} - 117 q^{11} - 444 q^{12} - 60 q^{13} + 128 q^{14} + 438 q^{15} + 626 q^{16} + 130 q^{17} + 32 q^{18} - 152 q^{19} - 554 q^{20} + 22 q^{21} - 1298 q^{22} - 254 q^{23} + 84 q^{24} - 314 q^{25} + 330 q^{26} + 130 q^{27} - 238 q^{28} + 38 q^{29} + 862 q^{30} + 1030 q^{31} + 172 q^{32} + 339 q^{33} + 916 q^{34} + 31 q^{35} - 110 q^{36} + 630 q^{37} - 814 q^{38} - 1644 q^{39} - 926 q^{40} - 2804 q^{41} - 816 q^{42} + 1600 q^{43} + 2174 q^{44} - 776 q^{45} - 2350 q^{46} - 798 q^{47} - 2258 q^{48} - 1309 q^{49} + 640 q^{50} + 712 q^{51} + 2774 q^{52} + 4870 q^{53} + 6450 q^{54} + 3378 q^{55} + 2934 q^{56} + 948 q^{57} - 2418 q^{58} - 4480 q^{59} - 10750 q^{60} - 5430 q^{61} - 14402 q^{62} - 10380 q^{63} - 12508 q^{64} - 7982 q^{65} - 10844 q^{66} - 2238 q^{67} - 1086 q^{68} + 5774 q^{69} + 6706 q^{70} + 7794 q^{71} + 19414 q^{72} + 10018 q^{73} + 11694 q^{74} + 18550 q^{75} + 21604 q^{76} + 8814 q^{77} + 21872 q^{78} + 6274 q^{79} + 24554 q^{80} + 13530 q^{81} + 13224 q^{82} + 226 q^{83} + 8578 q^{84} - 6390 q^{85} - 4600 q^{86} - 11778 q^{87} - 19220 q^{88} - 10858 q^{89} - 27138 q^{90} - 12329 q^{91} - 16632 q^{92} - 10134 q^{93} - 10502 q^{94} - 7074 q^{95} - 14394 q^{96} + 1210 q^{97} - 528 q^{98} - 3254 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(77))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
77.4.a $$\chi_{77}(1, \cdot)$$ 77.4.a.a 1 1
77.4.a.b 2
77.4.a.c 4
77.4.a.d 4
77.4.a.e 5
77.4.b $$\chi_{77}(76, \cdot)$$ 77.4.b.a 2 1
77.4.b.b 20
77.4.e $$\chi_{77}(23, \cdot)$$ 77.4.e.a 2 2
77.4.e.b 18
77.4.e.c 20
77.4.f $$\chi_{77}(15, \cdot)$$ 77.4.f.a 32 4
77.4.f.b 40
77.4.i $$\chi_{77}(10, \cdot)$$ 77.4.i.a 44 2
77.4.l $$\chi_{77}(6, \cdot)$$ 77.4.l.a 8 4
77.4.l.b 80
77.4.m $$\chi_{77}(4, \cdot)$$ 77.4.m.a 176 8
77.4.n $$\chi_{77}(17, \cdot)$$ 77.4.n.a 176 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(77))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(77)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 1}$$