## Defining parameters

 Level: $$N$$ = $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$2520$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(76))$$.

Total New Old
Modular forms 1125 640 485
Cusp forms 1035 608 427
Eisenstein series 90 32 58

## Trace form

 $$608q - 17q^{2} + 215q^{4} - 58q^{5} - 969q^{6} + 1399q^{8} + 906q^{9} + O(q^{10})$$ $$608q - 17q^{2} + 215q^{4} - 58q^{5} - 969q^{6} + 1399q^{8} + 906q^{9} - 89q^{10} - 3849q^{12} - 15962q^{13} + 9591q^{14} + 11664q^{15} - 8713q^{16} + 26246q^{17} + 1830q^{18} - 21168q^{19} + 2222q^{20} - 79242q^{21} - 29769q^{22} + 3960q^{23} + 46071q^{24} + 140706q^{25} - 11737q^{26} - 124830q^{27} + 77028q^{28} + 13910q^{29} + 333930q^{30} + 30780q^{31} - 236762q^{32} - 185754q^{33} - 275756q^{34} - 225432q^{35} - 388803q^{36} - 8012q^{37} + 380034q^{38} + 427140q^{39} + 664420q^{40} + 75134q^{41} + 695841q^{42} + 337176q^{43} - 249774q^{44} - 889554q^{45} - 1132464q^{46} - 886860q^{47} - 733476q^{48} - 148174q^{49} + 917802q^{50} + 1534410q^{51} + 328375q^{52} + 1484198q^{53} - 471528q^{54} + 318816q^{55} - 460818q^{56} - 894978q^{57} - 204002q^{58} - 1134000q^{59} + 1972902q^{60} - 2012126q^{61} + 233796q^{62} + 555840q^{63} - 3147889q^{64} + 2475562q^{65} - 899169q^{66} + 879552q^{67} + 231152q^{68} - 2243850q^{69} + 4591491q^{70} - 1528560q^{71} + 4956348q^{72} - 4328912q^{73} + 1145039q^{74} - 1065699q^{76} + 4866744q^{77} - 5598543q^{78} + 1866024q^{79} - 4344049q^{80} + 4065372q^{81} - 6378350q^{82} - 938448q^{83} + 1651755q^{84} - 2931874q^{85} + 2655726q^{86} + 1179216q^{87} + 8630991q^{88} - 4895422q^{89} + 6438846q^{90} + 1260864q^{91} - 3257094q^{92} + 4508394q^{93} - 14661870q^{94} - 6508080q^{95} - 1465050q^{96} + 7023670q^{97} + 12839854q^{98} + 1394874q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(76))$$

We only show spaces with odd 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
76.7.b $$\chi_{76}(39, \cdot)$$ 76.7.b.a 54 1
76.7.c $$\chi_{76}(37, \cdot)$$ 76.7.c.a 2 1
76.7.c.b 8
76.7.g $$\chi_{76}(7, \cdot)$$ 76.7.g.a 116 2
76.7.h $$\chi_{76}(65, \cdot)$$ 76.7.h.a 20 2
76.7.j $$\chi_{76}(13, \cdot)$$ 76.7.j.a 60 6
76.7.l $$\chi_{76}(23, \cdot)$$ 76.7.l.a 348 6

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(76))$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(\Gamma_1(76)) \cong$$ $$S_{7}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 2}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 2}$$