## Defining parameters

 Level: $$N$$ = $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$2880$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(76))$$.

Total New Old
Modular forms 1305 753 552
Cusp forms 1215 717 498
Eisenstein series 90 36 54

## Trace form

 $$717q - 9q^{2} - 9q^{4} - 18q^{5} - 9q^{6} - 9q^{8} - 18q^{9} + O(q^{10})$$ $$717q - 9q^{2} - 9q^{4} - 18q^{5} - 9q^{6} - 9q^{8} - 18q^{9} - 9q^{10} - 9q^{12} + 22146q^{13} - 9q^{14} - 61722q^{15} - 9q^{16} + 29349q^{17} + 100527q^{19} - 18q^{20} - 142335q^{21} - 9q^{22} - 98397q^{23} - 9q^{24} + 258300q^{25} - 9q^{26} + 845649q^{27} - 756738q^{28} - 597996q^{29} - 308772q^{30} + 424098q^{31} + 1640556q^{32} + 1644984q^{33} + 453546q^{34} - 583596q^{35} - 3515319q^{36} - 911754q^{37} - 2923020q^{38} - 1397034q^{39} + 436518q^{40} + 793260q^{41} + 6739821q^{42} + 1719429q^{43} + 3304296q^{44} + 5337261q^{45} - 2094714q^{46} - 1214253q^{47} - 11350530q^{48} - 4946025q^{49} + 2607084q^{50} + 817524q^{51} - 9q^{52} + 645354q^{53} - 19692q^{54} + 5345964q^{55} + 7469217q^{57} - 18q^{58} + 1825695q^{59} - 15228576q^{60} - 7236120q^{61} + 16542576q^{62} - 18814716q^{63} + 14544495q^{64} - 7442613q^{65} - 14099841q^{66} + 14941275q^{67} - 21818988q^{68} + 57358971q^{69} - 20398509q^{70} + 4319235q^{71} + 16435296q^{72} - 29130294q^{73} + 26838207q^{74} - 17718750q^{75} + 36422901q^{76} - 56619027q^{77} + 15920073q^{78} + 12511968q^{79} - 15381009q^{80} + 55291761q^{81} - 75428154q^{82} + 8066403q^{83} - 66131973q^{84} + 44757306q^{85} - 13190382q^{86} - 51590052q^{87} + 40974255q^{88} - 57553083q^{89} + 85996116q^{90} - 13484028q^{91} + 17289126q^{92} + 104933412q^{93} + 51790437q^{95} - 61682634q^{96} - 40890105q^{97} - 137750292q^{98} - 73473813q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(76))$$

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
76.8.a $$\chi_{76}(1, \cdot)$$ 76.8.a.a 5 1
76.8.a.b 6
76.8.d $$\chi_{76}(75, \cdot)$$ 76.8.d.a 68 1
76.8.e $$\chi_{76}(45, \cdot)$$ 76.8.e.a 22 2
76.8.f $$\chi_{76}(27, \cdot)$$ 76.8.f.a 136 2
76.8.i $$\chi_{76}(5, \cdot)$$ 76.8.i.a 72 6
76.8.k $$\chi_{76}(3, \cdot)$$ 76.8.k.a 408 6

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(76))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(76)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 2}$$