Properties

Label 75.9
Level 75
Weight 9
Dimension 1086
Nonzero newspaces 6
Newform subspaces 20
Sturm bound 3600
Trace bound 3

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 20 \)
Sturm bound: \(3600\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(75))\).

Total New Old
Modular forms 1656 1128 528
Cusp forms 1544 1086 458
Eisenstein series 112 42 70

Trace form

\( 1086 q - 56 q^{3} - 500 q^{4} + 444 q^{5} + 6526 q^{6} - 12912 q^{7} - 34920 q^{8} + 26616 q^{9} + O(q^{10}) \) \( 1086 q - 56 q^{3} - 500 q^{4} + 444 q^{5} + 6526 q^{6} - 12912 q^{7} - 34920 q^{8} + 26616 q^{9} + 38176 q^{10} + 47232 q^{11} + 29214 q^{12} - 310312 q^{13} + 105366 q^{15} - 272292 q^{16} - 405300 q^{17} + 23110 q^{18} - 824808 q^{19} - 641844 q^{20} + 1183398 q^{21} + 3191740 q^{22} + 437640 q^{23} - 3463652 q^{24} - 4269196 q^{25} - 2757984 q^{26} + 3798424 q^{27} + 10598908 q^{28} + 3913800 q^{29} + 533314 q^{30} - 2665336 q^{31} - 1282920 q^{32} - 2988150 q^{33} - 4410692 q^{34} + 853140 q^{35} + 5794302 q^{36} - 28302392 q^{37} + 9351420 q^{38} + 4347978 q^{39} + 8295032 q^{40} + 11981328 q^{41} - 10474070 q^{42} - 13446832 q^{43} + 4277700 q^{44} + 26903294 q^{45} + 1397188 q^{46} - 38400 q^{47} + 20633724 q^{48} - 13109266 q^{49} - 47100924 q^{50} - 15249996 q^{51} - 157666632 q^{52} - 113641080 q^{53} + 15398024 q^{54} + 45587884 q^{55} + 100370760 q^{56} + 119106938 q^{57} + 186341240 q^{58} + 95587200 q^{59} - 73576726 q^{60} + 28886976 q^{61} - 260903820 q^{62} - 161213512 q^{63} - 113529632 q^{64} + 24018972 q^{65} - 72153682 q^{66} - 13994832 q^{67} + 395623680 q^{68} + 156516484 q^{69} + 75974740 q^{70} + 171363936 q^{71} - 400407330 q^{72} + 195564728 q^{73} + 92323546 q^{75} - 80825560 q^{76} - 194351760 q^{77} - 201360160 q^{78} - 749072208 q^{79} - 339741204 q^{80} + 142507296 q^{81} - 160135380 q^{82} + 1070575200 q^{83} + 1303037558 q^{84} + 1181847688 q^{85} + 305840712 q^{86} - 160178620 q^{87} - 1108991540 q^{88} - 1141686000 q^{89} - 9890394 q^{90} - 893680540 q^{91} - 1059529800 q^{92} - 1032806612 q^{93} + 320194588 q^{94} + 1436174256 q^{95} + 485015522 q^{96} + 1749571208 q^{97} + 1127781240 q^{98} + 1095541740 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(75))\)

We only show spaces with odd 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
75.9.c \(\chi_{75}(26, \cdot)\) 75.9.c.a 1 1
75.9.c.b 1
75.9.c.c 2
75.9.c.d 2
75.9.c.e 10
75.9.c.f 10
75.9.c.g 10
75.9.c.h 12
75.9.d \(\chi_{75}(74, \cdot)\) 75.9.d.a 2 1
75.9.d.b 4
75.9.d.c 20
75.9.d.d 20
75.9.f \(\chi_{75}(7, \cdot)\) 75.9.f.a 4 2
75.9.f.b 8
75.9.f.c 8
75.9.f.d 12
75.9.f.e 16
75.9.h \(\chi_{75}(14, \cdot)\) 75.9.h.a 312 4
75.9.j \(\chi_{75}(11, \cdot)\) 75.9.j.a 312 4
75.9.k \(\chi_{75}(13, \cdot)\) 75.9.k.a 320 8

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)