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Properties

Label 72.12
Level 72
Weight 12
Dimension 698
Nonzero newspaces 6
Newform subspaces 18
Sturm bound 3456
Trace bound 2

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Defining parameters

Level: \( N \) = \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 18 \)
Sturm bound: \(3456\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(72))\).

Total New Old
Modular forms 1632 716 916
Cusp forms 1536 698 838
Eisenstein series 96 18 78

Trace form

\( 698 q + 20 q^{2} - 267 q^{3} + 434 q^{4} - 8122 q^{5} - 4100 q^{6} - 69210 q^{7} + 251162 q^{8} - 128163 q^{9} + O(q^{10}) \) \( 698 q + 20 q^{2} - 267 q^{3} + 434 q^{4} - 8122 q^{5} - 4100 q^{6} - 69210 q^{7} + 251162 q^{8} - 128163 q^{9} - 253520 q^{10} + 1120331 q^{11} - 56534 q^{12} + 3212 q^{13} + 228242 q^{14} - 1356456 q^{15} + 1187974 q^{16} - 2346106 q^{17} - 25031256 q^{18} + 30395214 q^{19} + 68879546 q^{20} - 20143116 q^{21} - 72188894 q^{22} + 99411750 q^{23} - 54553992 q^{24} - 263955921 q^{25} + 173291312 q^{26} + 230313600 q^{27} - 415405720 q^{28} - 343831524 q^{29} - 520353002 q^{30} - 146063680 q^{31} + 1879273510 q^{32} + 526847171 q^{33} - 254323466 q^{34} - 1855869084 q^{35} - 2357835262 q^{36} + 65776068 q^{37} + 4814463514 q^{38} + 2306560380 q^{39} - 2894646870 q^{40} - 485962285 q^{41} - 5223966358 q^{42} + 3581717245 q^{43} + 3087656178 q^{44} + 6735899726 q^{45} + 158432792 q^{46} - 8949682782 q^{47} - 15271033564 q^{48} - 10349528887 q^{49} + 6431485322 q^{50} + 15379276881 q^{51} + 11060361830 q^{52} + 5470160816 q^{53} - 17713337184 q^{54} - 17350004520 q^{55} - 23221374784 q^{56} - 21637032167 q^{57} + 24238564382 q^{58} + 55678698125 q^{59} - 14058012642 q^{60} + 18916914434 q^{61} + 25279540924 q^{62} - 56939912608 q^{63} - 7024807828 q^{64} - 27056814986 q^{65} - 68962885964 q^{66} - 22409938557 q^{67} + 60270213948 q^{68} - 192630974 q^{69} - 15004635282 q^{70} - 85319582432 q^{71} + 32620422522 q^{72} + 46358274738 q^{73} + 94980138230 q^{74} + 120879672961 q^{75} - 113551144410 q^{76} - 43139963724 q^{77} - 114851423750 q^{78} - 62230584892 q^{79} + 168974905440 q^{80} - 48295490551 q^{81} + 91826904792 q^{82} + 146925129172 q^{83} - 168439395688 q^{84} + 76294386724 q^{85} - 250290948574 q^{86} - 133584654822 q^{87} - 61423556786 q^{88} + 13881287516 q^{89} - 387119948930 q^{90} + 156540761784 q^{91} + 384316213446 q^{92} + 105083299394 q^{93} + 289770198324 q^{94} - 564762321328 q^{95} - 766352130432 q^{96} - 82075813461 q^{97} - 721294281294 q^{98} + 277947701886 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(72))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
72.12.a \(\chi_{72}(1, \cdot)\) 72.12.a.a 1 1
72.12.a.b 1
72.12.a.c 1
72.12.a.d 1
72.12.a.e 2
72.12.a.f 2
72.12.a.g 3
72.12.a.h 3
72.12.c \(\chi_{72}(71, \cdot)\) None 0 1
72.12.d \(\chi_{72}(37, \cdot)\) 72.12.d.a 2 1
72.12.d.b 10
72.12.d.c 20
72.12.d.d 22
72.12.f \(\chi_{72}(35, \cdot)\) 72.12.f.a 44 1
72.12.i \(\chi_{72}(25, \cdot)\) 72.12.i.a 32 2
72.12.i.b 34
72.12.l \(\chi_{72}(11, \cdot)\) 72.12.l.a 4 2
72.12.l.b 256
72.12.n \(\chi_{72}(13, \cdot)\) 72.12.n.a 260 2
72.12.o \(\chi_{72}(23, \cdot)\) None 0 2

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(72))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(72)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 1}\)