## Defining parameters

 Level: $$N$$ = $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$1080$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(76))$$.

Total New Old
Modular forms 405 224 181
Cusp forms 315 192 123
Eisenstein series 90 32 58

## Trace form

 $$192q - 9q^{2} - 9q^{4} - 18q^{5} - 9q^{6} - 9q^{8} - 18q^{9} + O(q^{10})$$ $$192q - 9q^{2} - 9q^{4} - 18q^{5} - 9q^{6} - 9q^{8} - 18q^{9} - 9q^{10} - 9q^{12} + 42q^{13} - 9q^{14} + 54q^{15} - 9q^{16} - 9q^{17} - 18q^{18} - 21q^{19} - 18q^{20} - 81q^{21} - 9q^{22} - 45q^{23} - 9q^{24} - 144q^{25} - 9q^{26} - 198q^{27} - 252q^{28} - 162q^{29} - 450q^{30} - 108q^{31} - 234q^{32} - 234q^{33} - 144q^{34} - 72q^{35} - 99q^{36} - 36q^{37} + 54q^{38} + 108q^{39} + 180q^{40} + 54q^{41} + 441q^{42} + 255q^{43} + 306q^{44} + 711q^{45} + 396q^{46} + 423q^{47} + 684q^{48} + 345q^{49} + 342q^{50} + 351q^{51} - 9q^{52} + 90q^{53} + 72q^{54} + 36q^{55} - 18q^{56} - 63q^{57} - 18q^{58} - 135q^{59} + 342q^{60} - 786q^{61} + 576q^{62} - 540q^{63} + 783q^{64} - 963q^{65} + 639q^{66} - 519q^{67} + 432q^{68} - 1233q^{69} + 531q^{70} - 477q^{71} + 396q^{72} - 231q^{73} + 27q^{74} - 99q^{76} + 369q^{77} - 243q^{78} + 438q^{79} - 369q^{80} + 936q^{81} - 954q^{82} + 441q^{83} - 981q^{84} + 1746q^{85} - 702q^{86} + 936q^{87} - 945q^{88} + 873q^{89} - 1314q^{90} + 492q^{91} - 774q^{92} + 810q^{93} - 702q^{94} - 675q^{95} - 1242q^{96} - 675q^{97} - 1494q^{98} - 1242q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(76))$$

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
76.3.b $$\chi_{76}(39, \cdot)$$ 76.3.b.a 4 1
76.3.b.b 14
76.3.c $$\chi_{76}(37, \cdot)$$ 76.3.c.a 2 1
76.3.c.b 2
76.3.g $$\chi_{76}(7, \cdot)$$ 76.3.g.a 4 2
76.3.g.b 4
76.3.g.c 28
76.3.h $$\chi_{76}(65, \cdot)$$ 76.3.h.a 8 2
76.3.j $$\chi_{76}(13, \cdot)$$ 76.3.j.a 18 6
76.3.l $$\chi_{76}(23, \cdot)$$ 76.3.l.a 108 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(76))$$ into lower level spaces

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