Properties

Label 40.3
Level 40
Weight 3
Dimension 44
Nonzero newspaces 4
Newform subspaces 7
Sturm bound 288
Trace bound 1

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Defining parameters

Level: \( N \) = \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
Sturm bound: \(288\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(40))\).

Total New Old
Modular forms 120 56 64
Cusp forms 72 44 28
Eisenstein series 48 12 36

Trace form

\( 44 q - 12 q^{4} + 4 q^{5} - 16 q^{6} + 4 q^{7} + 12 q^{8} + 2 q^{9} + O(q^{10}) \) \( 44 q - 12 q^{4} + 4 q^{5} - 16 q^{6} + 4 q^{7} + 12 q^{8} + 2 q^{9} - 4 q^{10} - 44 q^{11} - 24 q^{12} - 38 q^{13} - 64 q^{14} - 76 q^{15} - 88 q^{16} - 26 q^{17} - 52 q^{18} + 60 q^{19} + 16 q^{20} + 56 q^{21} + 112 q^{22} + 92 q^{23} + 104 q^{24} - 8 q^{25} + 104 q^{26} + 24 q^{27} + 168 q^{28} + 240 q^{30} - 152 q^{31} + 200 q^{32} - 72 q^{33} + 164 q^{34} - 172 q^{35} + 128 q^{36} - 82 q^{37} - 128 q^{38} - 44 q^{40} + 132 q^{41} - 144 q^{42} + 144 q^{43} - 320 q^{44} + 154 q^{45} - 336 q^{46} + 60 q^{47} - 536 q^{48} - 106 q^{49} - 484 q^{50} - 48 q^{51} - 492 q^{52} - 66 q^{53} - 296 q^{54} + 8 q^{55} - 176 q^{56} - 296 q^{57} - 52 q^{58} + 156 q^{59} + 120 q^{60} - 24 q^{61} + 176 q^{62} + 356 q^{63} + 288 q^{64} + 14 q^{65} + 688 q^{66} + 384 q^{67} + 756 q^{68} + 760 q^{70} + 520 q^{71} + 820 q^{72} + 202 q^{73} + 636 q^{74} + 584 q^{75} + 368 q^{76} + 72 q^{77} + 232 q^{78} + 56 q^{80} + 24 q^{81} - 596 q^{82} - 664 q^{83} - 800 q^{84} - 198 q^{85} - 800 q^{86} - 808 q^{87} - 704 q^{88} + 100 q^{89} - 1204 q^{90} - 744 q^{91} - 688 q^{92} + 120 q^{93} - 744 q^{94} - 592 q^{95} - 944 q^{96} + 362 q^{97} + 104 q^{98} - 516 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(40))\)

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
40.3.b \(\chi_{40}(31, \cdot)\) None 0 1
40.3.e \(\chi_{40}(19, \cdot)\) 40.3.e.a 1 1
40.3.e.b 1
40.3.e.c 8
40.3.g \(\chi_{40}(11, \cdot)\) 40.3.g.a 8 1
40.3.h \(\chi_{40}(39, \cdot)\) None 0 1
40.3.i \(\chi_{40}(13, \cdot)\) 40.3.i.a 20 2
40.3.l \(\chi_{40}(17, \cdot)\) 40.3.l.a 2 2
40.3.l.b 4

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(40)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)