Properties

Label 72.8
Level 72
Weight 8
Dimension 441
Nonzero newspaces 6
Newform subspaces 17
Sturm bound 2304
Trace bound 2

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Defining parameters

Level: \( N \) = \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 17 \)
Sturm bound: \(2304\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(72))\).

Total New Old
Modular forms 1056 459 597
Cusp forms 960 441 519
Eisenstein series 96 18 78

Trace form

\( 441 q + 4 q^{2} + 9 q^{3} + 114 q^{4} - 152 q^{5} - 260 q^{6} + 990 q^{7} - 3686 q^{8} - 2439 q^{9} + O(q^{10}) \) \( 441 q + 4 q^{2} + 9 q^{3} + 114 q^{4} - 152 q^{5} - 260 q^{6} + 990 q^{7} - 3686 q^{8} - 2439 q^{9} + 5520 q^{10} + 10459 q^{11} + 9226 q^{12} - 5226 q^{13} - 45422 q^{14} - 30216 q^{15} + 10374 q^{16} + 65596 q^{17} + 75528 q^{18} - 139590 q^{19} - 5894 q^{20} - 96276 q^{21} - 36126 q^{22} + 372846 q^{23} - 159048 q^{24} - 111156 q^{25} - 88592 q^{26} - 765936 q^{27} + 169320 q^{28} - 166050 q^{29} + 887638 q^{30} + 302928 q^{31} - 45466 q^{32} + 71543 q^{33} + 301974 q^{34} - 5484 q^{35} - 1347070 q^{36} + 104862 q^{37} - 768166 q^{38} + 292500 q^{39} + 1025130 q^{40} - 1950227 q^{41} + 3812042 q^{42} + 1458189 q^{43} - 1962318 q^{44} - 1163194 q^{45} - 2978088 q^{46} - 1798566 q^{47} + 2374148 q^{48} + 1903524 q^{49} + 265402 q^{50} + 1087077 q^{51} - 1372122 q^{52} - 4558862 q^{53} + 425472 q^{54} - 532080 q^{55} - 821888 q^{56} + 4147405 q^{57} + 2352318 q^{58} - 4835483 q^{59} + 10462878 q^{60} + 584388 q^{61} + 1225532 q^{62} + 8835008 q^{63} - 2360340 q^{64} - 5513806 q^{65} - 19994252 q^{66} - 8078133 q^{67} - 5211780 q^{68} - 5179742 q^{69} + 8000238 q^{70} + 5416616 q^{71} + 33011610 q^{72} + 11514744 q^{73} + 1361686 q^{74} - 2550059 q^{75} - 23928090 q^{76} - 9392796 q^{77} - 22352774 q^{78} - 517524 q^{79} - 36426720 q^{80} - 3760819 q^{81} + 14171640 q^{82} + 16607168 q^{83} + 50755928 q^{84} + 15961824 q^{85} + 69215458 q^{86} + 11076978 q^{87} + 22477902 q^{88} + 29050450 q^{89} - 41056610 q^{90} - 42548856 q^{91} - 98967546 q^{92} - 18593230 q^{93} - 53043756 q^{94} + 8042272 q^{95} + 12197472 q^{96} + 2512893 q^{97} + 57779586 q^{98} + 9115374 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(72))\)

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
72.8.a \(\chi_{72}(1, \cdot)\) 72.8.a.a 1 1
72.8.a.b 1
72.8.a.c 1
72.8.a.d 1
72.8.a.e 1
72.8.a.f 2
72.8.a.g 2
72.8.c \(\chi_{72}(71, \cdot)\) None 0 1
72.8.d \(\chi_{72}(37, \cdot)\) 72.8.d.a 2 1
72.8.d.b 6
72.8.d.c 12
72.8.d.d 14
72.8.f \(\chi_{72}(35, \cdot)\) 72.8.f.a 28 1
72.8.i \(\chi_{72}(25, \cdot)\) 72.8.i.a 20 2
72.8.i.b 22
72.8.l \(\chi_{72}(11, \cdot)\) 72.8.l.a 4 2
72.8.l.b 160
72.8.n \(\chi_{72}(13, \cdot)\) 72.8.n.a 164 2
72.8.o \(\chi_{72}(23, \cdot)\) None 0 2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(72))\) into lower level spaces

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