## Defining parameters

 Level: $$N$$ = $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$14$$ Sturm bound: $$1152$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(72))$$.

Total New Old
Modular forms 480 202 278
Cusp forms 384 184 200
Eisenstein series 96 18 78

## Trace form

 $$184q - 4q^{2} - 3q^{3} - 14q^{4} - 22q^{5} - 20q^{6} - 42q^{7} + 26q^{8} + 21q^{9} + O(q^{10})$$ $$184q - 4q^{2} - 3q^{3} - 14q^{4} - 22q^{5} - 20q^{6} - 42q^{7} + 26q^{8} + 21q^{9} + 80q^{10} + 107q^{11} + 106q^{12} + 64q^{13} + 146q^{14} + 24q^{15} + 134q^{16} - 142q^{17} - 120q^{18} - 186q^{19} - 454q^{20} + 228q^{21} - 622q^{22} + 246q^{23} - 264q^{24} + 9q^{25} + 32q^{26} + 288q^{27} - 152q^{28} - 192q^{29} + 118q^{30} - 32q^{31} + 166q^{32} - 469q^{33} + 1382q^{34} - 444q^{35} - 766q^{36} - 168q^{37} + 394q^{38} - 276q^{39} + 810q^{40} - 697q^{41} - 118q^{42} - 1507q^{43} - 1038q^{44} + 446q^{45} - 2696q^{46} - 1326q^{47} - 796q^{48} + 703q^{49} - 3118q^{50} + 345q^{51} - 1562q^{52} + 1940q^{53} + 624q^{54} + 1800q^{55} + 2624q^{56} + 1537q^{57} + 4798q^{58} + 4349q^{59} + 798q^{60} - 602q^{61} + 4300q^{62} + 1184q^{63} + 1900q^{64} - 146q^{65} + 2500q^{66} - 813q^{67} - 1860q^{68} - 3710q^{69} - 6402q^{70} - 6032q^{71} + 954q^{72} - 1410q^{73} - 1498q^{74} - 5879q^{75} - 1242q^{76} - 1260q^{77} + 3946q^{78} + 3796q^{79} + 8160q^{80} + 1937q^{81} + 8904q^{82} + 7372q^{83} + 152q^{84} + 1244q^{85} + 146q^{86} + 7050q^{87} + 4430q^{88} + 4616q^{89} + 1870q^{90} - 2088q^{91} - 954q^{92} + 3746q^{93} - 5052q^{94} - 5008q^{95} - 4032q^{96} - 561q^{97} - 10326q^{98} - 7458q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$

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
72.4.a $$\chi_{72}(1, \cdot)$$ 72.4.a.a 1 1
72.4.a.b 1
72.4.a.c 1
72.4.a.d 1
72.4.c $$\chi_{72}(71, \cdot)$$ None 0 1
72.4.d $$\chi_{72}(37, \cdot)$$ 72.4.d.a 2 1
72.4.d.b 2
72.4.d.c 4
72.4.d.d 6
72.4.f $$\chi_{72}(35, \cdot)$$ 72.4.f.a 12 1
72.4.i $$\chi_{72}(25, \cdot)$$ 72.4.i.a 8 2
72.4.i.b 10
72.4.l $$\chi_{72}(11, \cdot)$$ 72.4.l.a 4 2
72.4.l.b 64
72.4.n $$\chi_{72}(13, \cdot)$$ 72.4.n.a 68 2
72.4.o $$\chi_{72}(23, \cdot)$$ None 0 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(72))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(72)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$