## 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

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

## 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

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