## Defining parameters

 Level: $$N$$ = $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$11$$ Sturm bound: $$1008$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(78))$$.

Total New Old
Modular forms 384 84 300
Cusp forms 288 84 204
Eisenstein series 96 0 96

## Trace form

 $$84 q + 40 q^{7} + 24 q^{8} + 12 q^{9} + O(q^{10})$$ $$84 q + 40 q^{7} + 24 q^{8} + 12 q^{9} + 60 q^{10} + 24 q^{11} - 48 q^{13} - 48 q^{14} - 72 q^{15} - 32 q^{16} - 72 q^{17} - 36 q^{18} - 80 q^{19} - 48 q^{20} + 60 q^{21} + 48 q^{23} - 72 q^{27} - 120 q^{29} - 192 q^{30} - 144 q^{31} - 372 q^{33} - 240 q^{35} - 48 q^{36} - 72 q^{37} + 96 q^{39} + 60 q^{41} + 144 q^{42} + 256 q^{43} + 456 q^{45} + 96 q^{47} + 16 q^{49} + 84 q^{50} + 624 q^{51} + 72 q^{52} + 480 q^{53} + 360 q^{54} + 456 q^{55} + 192 q^{56} + 396 q^{57} + 324 q^{58} + 168 q^{59} + 96 q^{60} - 156 q^{61} + 144 q^{62} - 300 q^{63} - 180 q^{65} - 96 q^{66} - 416 q^{67} - 24 q^{68} - 708 q^{69} - 432 q^{70} - 336 q^{71} - 192 q^{72} - 512 q^{73} - 156 q^{74} - 360 q^{75} + 160 q^{76} - 216 q^{78} + 336 q^{79} - 96 q^{80} - 180 q^{81} + 660 q^{82} + 504 q^{83} + 168 q^{84} + 828 q^{85} + 192 q^{86} + 696 q^{87} + 432 q^{89} - 104 q^{91} - 96 q^{92} + 660 q^{93} - 480 q^{94} - 480 q^{95} - 1320 q^{97} - 384 q^{98} - 612 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(78))$$

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
78.3.c $$\chi_{78}(53, \cdot)$$ 78.3.c.a 8 1
78.3.d $$\chi_{78}(77, \cdot)$$ 78.3.d.a 4 1
78.3.d.b 4
78.3.f $$\chi_{78}(31, \cdot)$$ 78.3.f.a 4 2
78.3.f.b 8
78.3.h $$\chi_{78}(29, \cdot)$$ 78.3.h.a 8 2
78.3.h.b 12
78.3.j $$\chi_{78}(17, \cdot)$$ 78.3.j.a 20 2
78.3.l $$\chi_{78}(7, \cdot)$$ 78.3.l.a 4 4
78.3.l.b 4
78.3.l.c 8

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(78))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(78)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$