## Defining parameters

 Level: $$N$$ = $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ = $$6$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$2160$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_1(76))$$.

Total New Old
Modular forms 945 541 404
Cusp forms 855 505 350
Eisenstein series 90 36 54

## Trace form

 $$505q - 9q^{2} + 24q^{3} - 9q^{4} - 126q^{5} - 9q^{6} + 176q^{7} - 9q^{8} + 180q^{9} + O(q^{10})$$ $$505q - 9q^{2} + 24q^{3} - 9q^{4} - 126q^{5} - 9q^{6} + 176q^{7} - 9q^{8} + 180q^{9} - 9q^{10} - 1080q^{11} - 9q^{12} + 2990q^{13} - 9q^{14} + 3078q^{15} - 9q^{16} - 3357q^{17} - 7241q^{19} - 18q^{20} - 1185q^{21} - 9q^{22} + 13347q^{23} - 9q^{24} + 12082q^{25} - 9q^{26} + 6705q^{27} + 14598q^{28} - 10188q^{29} - 69876q^{30} - 28150q^{31} - 33804q^{32} + 36q^{33} + 14346q^{34} + 39060q^{35} + 107001q^{36} + 25238q^{37} + 83844q^{38} + 47694q^{39} + 37854q^{40} - 35028q^{41} - 68499q^{42} - 55591q^{43} - 95904q^{44} - 134289q^{45} - 112554q^{46} + 40131q^{47} - 30186q^{48} + 103971q^{49} + 154188q^{50} + 140472q^{51} - 9q^{52} - 91530q^{53} - 2196q^{54} - 161316q^{55} - 128541q^{57} - 18q^{58} - 73269q^{59} - 244872q^{60} + 341000q^{61} - 43704q^{62} + 26388q^{63} + 323631q^{64} - 14319q^{65} + 368703q^{66} - 88657q^{67} + 187164q^{68} - 254547q^{69} - 29709q^{70} - 81261q^{71} - 448344q^{72} + 150170q^{73} - 378081q^{74} + 231234q^{75} - 499419q^{76} + 523431q^{77} - 365031q^{78} + 203696q^{79} - 73809q^{80} + 192213q^{81} + 395406q^{82} + 7623q^{83} + 733851q^{84} - 1008486q^{85} + 578394q^{86} - 145332q^{87} + 598383q^{88} - 433773q^{89} + 173916q^{90} - 191852q^{91} - 549954q^{92} + 475032q^{93} + 498645q^{95} + 684342q^{96} + 75425q^{97} - 586188q^{98} - 738837q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_1(76))$$

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
76.6.a $$\chi_{76}(1, \cdot)$$ 76.6.a.a 3 1
76.6.a.b 4
76.6.d $$\chi_{76}(75, \cdot)$$ 76.6.d.a 48 1
76.6.e $$\chi_{76}(45, \cdot)$$ 76.6.e.a 18 2
76.6.f $$\chi_{76}(27, \cdot)$$ 76.6.f.a 96 2
76.6.i $$\chi_{76}(5, \cdot)$$ 76.6.i.a 48 6
76.6.k $$\chi_{76}(3, \cdot)$$ 76.6.k.a 288 6

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_1(76))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_1(76)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 2}$$