## Defining parameters

 Level: $$N$$ = $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$864$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(72))$$.

Total New Old
Modular forms 336 137 199
Cusp forms 240 119 121
Eisenstein series 96 18 78

## Trace form

 $$119q - 4q^{2} - 6q^{3} + 2q^{4} + 4q^{6} + 22q^{7} + 26q^{8} + 6q^{9} + O(q^{10})$$ $$119q - 4q^{2} - 6q^{3} + 2q^{4} + 4q^{6} + 22q^{7} + 26q^{8} + 6q^{9} + 8q^{10} + 34q^{11} - 14q^{12} - 16q^{13} - 78q^{14} + 6q^{15} - 122q^{16} - 2q^{17} - 96q^{18} + 2q^{19} - 150q^{20} - 36q^{21} - 38q^{22} - 78q^{23} - 24q^{24} + q^{25} + 24q^{26} - 180q^{27} + 168q^{28} - 108q^{29} - 74q^{30} - 114q^{31} + 86q^{32} - 22q^{33} + 102q^{34} - 204q^{35} + 2q^{36} + 60q^{37} - 110q^{38} + 6q^{39} - 326q^{40} + 172q^{41} + 194q^{42} - 150q^{43} + 130q^{44} + 260q^{45} - 40q^{46} + 318q^{47} + 332q^{48} + 153q^{49} + 530q^{50} + 210q^{51} + 374q^{52} + 288q^{54} + 404q^{55} + 624q^{56} - 290q^{57} + 294q^{58} - 62q^{59} + 606q^{60} - 76q^{61} + 180q^{62} - 490q^{63} - 436q^{64} - 636q^{65} + 172q^{66} - 94q^{67} - 20q^{68} - 104q^{69} - 58q^{70} - 78q^{72} - 354q^{73} + 6q^{74} + 454q^{75} + 182q^{76} + 504q^{77} - 446q^{78} - 246q^{79} - 624q^{80} + 398q^{81} + 8q^{82} + 100q^{83} - 952q^{84} - 188q^{85} - 998q^{86} + 138q^{87} - 626q^{88} - 50q^{89} - 1250q^{90} - 780q^{91} - 1098q^{92} - 352q^{93} - 828q^{94} - 684q^{95} - 888q^{96} - 252q^{97} - 1078q^{98} - 510q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(72))$$

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
72.3.b $$\chi_{72}(19, \cdot)$$ 72.3.b.a 1 1
72.3.b.b 4
72.3.b.c 4
72.3.e $$\chi_{72}(17, \cdot)$$ 72.3.e.a 2 1
72.3.g $$\chi_{72}(55, \cdot)$$ None 0 1
72.3.h $$\chi_{72}(53, \cdot)$$ 72.3.h.a 8 1
72.3.j $$\chi_{72}(5, \cdot)$$ 72.3.j.a 44 2
72.3.k $$\chi_{72}(7, \cdot)$$ None 0 2
72.3.m $$\chi_{72}(41, \cdot)$$ 72.3.m.a 4 2
72.3.m.b 8
72.3.p $$\chi_{72}(43, \cdot)$$ 72.3.p.a 4 2
72.3.p.b 40

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(72))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(72)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$