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Properties

Label 75.12
Level 75
Weight 12
Dimension 1499
Nonzero newspaces 6
Newform subspaces 27
Sturm bound 4800
Trace bound 1

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 27 \)
Sturm bound: \(4800\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(75))\).

Total New Old
Modular forms 2256 1539 717
Cusp forms 2144 1499 645
Eisenstein series 112 40 72

Trace form

\( 1499 q - 50 q^{2} - 1261 q^{3} + 16304 q^{4} + 2814 q^{5} + 19860 q^{6} - 340 q^{7} - 537504 q^{8} + 295235 q^{9} + O(q^{10}) \) \( 1499 q - 50 q^{2} - 1261 q^{3} + 16304 q^{4} + 2814 q^{5} + 19860 q^{6} - 340 q^{7} - 537504 q^{8} + 295235 q^{9} - 38464 q^{10} - 1995140 q^{11} + 3384626 q^{12} + 8062666 q^{13} - 8887728 q^{14} + 1519346 q^{15} + 56083660 q^{16} - 49024186 q^{17} + 8484980 q^{18} - 6038632 q^{19} - 54371844 q^{20} - 45449574 q^{21} + 33118308 q^{22} - 43862944 q^{23} - 324335016 q^{24} - 6429286 q^{25} - 445722636 q^{26} - 240914341 q^{27} + 1211301948 q^{28} + 1052181646 q^{29} - 1591003966 q^{30} - 1711172660 q^{31} + 1619935288 q^{32} + 2713616846 q^{33} + 980996488 q^{34} - 2616926020 q^{35} - 5088039214 q^{36} - 1310932720 q^{37} - 6801961028 q^{38} + 6952476396 q^{39} + 15061994712 q^{40} + 2241128746 q^{41} - 6762335886 q^{42} - 18425016952 q^{43} - 11258330340 q^{44} + 10664900644 q^{45} + 32502166820 q^{46} + 18558271232 q^{47} + 6244954048 q^{48} - 34707084653 q^{49} - 50993085244 q^{50} - 16231813222 q^{51} + 39938552776 q^{52} + 22844314636 q^{53} + 52035726066 q^{54} + 30399958724 q^{55} + 40858797240 q^{56} - 48419813990 q^{57} - 172940562876 q^{58} - 32062899508 q^{59} + 99019159994 q^{60} + 53468695122 q^{61} + 90177771596 q^{62} + 2382599836 q^{63} - 35882211240 q^{64} + 71329943122 q^{65} - 107893134378 q^{66} - 125297665504 q^{67} - 4603020040 q^{68} - 8358858308 q^{69} + 126003219460 q^{70} + 43823347544 q^{71} + 227360395494 q^{72} - 171104749970 q^{73} - 185742846508 q^{74} + 48734742486 q^{75} + 15954277752 q^{76} - 25870286784 q^{77} + 25182951784 q^{78} + 213956902180 q^{79} + 63547686676 q^{80} - 151495840981 q^{81} - 656494540504 q^{82} - 554968106780 q^{83} + 876427430918 q^{84} + 981163022978 q^{85} + 341413866640 q^{86} - 485546542854 q^{87} - 1558484661524 q^{88} - 494027242092 q^{89} - 214814580994 q^{90} - 12801343940 q^{91} + 989676512008 q^{92} + 122936589592 q^{93} + 1737437758396 q^{94} + 516413227576 q^{95} + 1397263850610 q^{96} + 422837261246 q^{97} - 1438130338682 q^{98} - 193760318052 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(75))\)

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
75.12.a \(\chi_{75}(1, \cdot)\) 75.12.a.a 1 1
75.12.a.b 1
75.12.a.c 2
75.12.a.d 2
75.12.a.e 3
75.12.a.f 3
75.12.a.g 3
75.12.a.h 4
75.12.a.i 4
75.12.a.j 6
75.12.a.k 6
75.12.b \(\chi_{75}(49, \cdot)\) 75.12.b.a 2 1
75.12.b.b 2
75.12.b.c 4
75.12.b.d 4
75.12.b.e 6
75.12.b.f 6
75.12.b.g 8
75.12.e \(\chi_{75}(32, \cdot)\) 75.12.e.a 4 2
75.12.e.b 4
75.12.e.c 24
75.12.e.d 40
75.12.e.e 56
75.12.g \(\chi_{75}(16, \cdot)\) 75.12.g.a 108 4
75.12.g.b 108
75.12.i \(\chi_{75}(4, \cdot)\) 75.12.i.a 224 4
75.12.l \(\chi_{75}(2, \cdot)\) 75.12.l.a 864 8

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)