## Defining parameters

 Level: $$N$$ = $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$576$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(72))$$.

Total New Old
Modular forms 192 73 119
Cusp forms 97 55 42
Eisenstein series 95 18 77

## Trace form

 $$55 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{5} - 8 q^{6} - 2 q^{7} - 14 q^{8} - 9 q^{9} + O(q^{10})$$ $$55 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + 4 q^{5} - 8 q^{6} - 2 q^{7} - 14 q^{8} - 9 q^{9} - 24 q^{10} - 17 q^{11} - 14 q^{12} - 2 q^{13} - 14 q^{14} - 24 q^{15} - 10 q^{16} - 20 q^{17} - 22 q^{19} + 10 q^{20} - 12 q^{21} + 14 q^{22} - 18 q^{23} + 24 q^{24} - 8 q^{25} + 28 q^{26} + 8 q^{28} + 6 q^{29} + 46 q^{30} - 16 q^{31} + 38 q^{32} - 7 q^{33} - 14 q^{34} + 36 q^{35} + 50 q^{36} + 6 q^{37} + 38 q^{38} + 36 q^{39} + 10 q^{40} - 5 q^{41} + 50 q^{42} + 25 q^{43} + 42 q^{44} + 2 q^{45} + 16 q^{46} + 42 q^{47} + 20 q^{48} - 52 q^{49} + 28 q^{50} + 9 q^{51} - 26 q^{52} - 14 q^{53} - 12 q^{54} - 16 q^{55} - 32 q^{56} - 29 q^{57} - 10 q^{58} + q^{59} - 66 q^{60} - 8 q^{61} - 64 q^{62} + 32 q^{63} - 20 q^{64} - 34 q^{65} - 104 q^{66} - 25 q^{67} - 84 q^{68} + 22 q^{69} - 46 q^{70} + 8 q^{71} - 102 q^{72} - 24 q^{73} - 110 q^{74} + 49 q^{75} - 34 q^{76} + 12 q^{77} - 74 q^{78} - 4 q^{79} - 96 q^{80} + 11 q^{81} + 20 q^{82} + 32 q^{83} - 64 q^{84} + 8 q^{85} - 50 q^{86} + 18 q^{87} + 14 q^{88} + 22 q^{89} - 14 q^{90} + 24 q^{91} + 54 q^{92} - 10 q^{93} + 24 q^{94} + 64 q^{95} + 48 q^{96} + 51 q^{97} + 96 q^{98} + 30 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(72))$$

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
72.2.a $$\chi_{72}(1, \cdot)$$ 72.2.a.a 1 1
72.2.c $$\chi_{72}(71, \cdot)$$ None 0 1
72.2.d $$\chi_{72}(37, \cdot)$$ 72.2.d.a 2 1
72.2.d.b 2
72.2.f $$\chi_{72}(35, \cdot)$$ 72.2.f.a 4 1
72.2.i $$\chi_{72}(25, \cdot)$$ 72.2.i.a 2 2
72.2.i.b 4
72.2.l $$\chi_{72}(11, \cdot)$$ 72.2.l.a 4 2
72.2.l.b 16
72.2.n $$\chi_{72}(13, \cdot)$$ 72.2.n.a 4 2
72.2.n.b 16
72.2.o $$\chi_{72}(23, \cdot)$$ None 0 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(72))$$ into lower level spaces

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