# Properties

 Label 72.4 Level 72 Weight 4 Dimension 184 Nonzero newspaces 6 Newform subspaces 14 Sturm bound 1152 Trace bound 2

## 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

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

## 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

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