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Properties

Label 75.7
Level 75
Weight 7
Dimension 809
Nonzero newspaces 6
Newform subspaces 16
Sturm bound 2800
Trace bound 3

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 16 \)
Sturm bound: \(2800\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(75))\).

Total New Old
Modular forms 1256 851 405
Cusp forms 1144 809 335
Eisenstein series 112 42 70

Trace form

\( 809 q - 32 q^{2} + 31 q^{3} + 60 q^{4} - 136 q^{5} - 1106 q^{6} + 1910 q^{7} + 6936 q^{8} + 2007 q^{9} + O(q^{10}) \) \( 809 q - 32 q^{2} + 31 q^{3} + 60 q^{4} - 136 q^{5} - 1106 q^{6} + 1910 q^{7} + 6936 q^{8} + 2007 q^{9} - 6064 q^{10} - 6496 q^{11} - 11154 q^{12} + 1046 q^{13} + 7296 q^{15} + 2012 q^{16} + 20040 q^{17} + 41494 q^{18} + 12454 q^{19} - 61844 q^{20} - 72912 q^{21} - 196580 q^{22} - 83392 q^{23} + 45916 q^{24} + 133364 q^{25} + 81312 q^{26} - 83321 q^{27} + 209564 q^{28} + 49600 q^{29} - 25166 q^{30} - 165378 q^{31} - 320488 q^{32} - 10458 q^{33} + 197996 q^{34} + 362080 q^{35} - 224082 q^{36} - 370 q^{37} - 271908 q^{38} - 188304 q^{39} + 753912 q^{40} - 356384 q^{41} + 87946 q^{42} - 833362 q^{43} - 1423100 q^{44} - 817606 q^{45} - 137660 q^{46} - 449056 q^{47} + 1218300 q^{48} + 2051643 q^{49} + 1983956 q^{50} + 1134900 q^{51} + 1726168 q^{52} - 3632 q^{53} - 852472 q^{54} + 607324 q^{55} - 1224120 q^{56} - 1571248 q^{57} - 4501256 q^{58} - 2525600 q^{59} - 1787206 q^{60} - 483834 q^{61} + 554644 q^{62} + 1499168 q^{63} + 10355296 q^{64} + 2707372 q^{65} + 1366286 q^{66} - 817282 q^{67} - 1523392 q^{68} - 2093042 q^{69} - 3729740 q^{70} - 1663168 q^{71} - 1520418 q^{72} - 3017890 q^{73} + 1384336 q^{75} + 6613768 q^{76} + 6411104 q^{77} + 5393840 q^{78} + 4849294 q^{79} + 1124876 q^{80} - 3547281 q^{81} - 8945876 q^{82} - 4038528 q^{83} - 4872874 q^{84} + 5992328 q^{85} - 4823416 q^{86} + 129638 q^{87} + 1033036 q^{88} + 2755500 q^{89} + 7656006 q^{90} + 8247584 q^{91} + 518712 q^{92} - 363704 q^{93} - 7812964 q^{94} - 9134624 q^{95} - 3220030 q^{96} - 11473282 q^{97} - 20730984 q^{98} - 10850100 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(75))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
75.7.c \(\chi_{75}(26, \cdot)\) 75.7.c.a 1 1
75.7.c.b 2
75.7.c.c 8
75.7.c.d 8
75.7.c.e 8
75.7.c.f 8
75.7.d \(\chi_{75}(74, \cdot)\) 75.7.d.a 2 1
75.7.d.b 16
75.7.d.c 16
75.7.f \(\chi_{75}(7, \cdot)\) 75.7.f.a 8 2
75.7.f.b 8
75.7.f.c 8
75.7.f.d 12
75.7.h \(\chi_{75}(14, \cdot)\) 75.7.h.a 232 4
75.7.j \(\chi_{75}(11, \cdot)\) 75.7.j.a 232 4
75.7.k \(\chi_{75}(13, \cdot)\) 75.7.k.a 240 8

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(75))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(75)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)