Properties

Label 13.9
Level 13
Weight 9
Dimension 50
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 126
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(126\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(13))\).

Total New Old
Modular forms 62 62 0
Cusp forms 50 50 0
Eisenstein series 12 12 0

Trace form

\( 50 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 6 q^{5} - 6 q^{6} + 2524 q^{7} - 6 q^{8} + 17490 q^{9} + O(q^{10}) \) \( 50 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 6 q^{5} - 6 q^{6} + 2524 q^{7} - 6 q^{8} + 17490 q^{9} - 26886 q^{10} - 25080 q^{11} + 86094 q^{13} + 105972 q^{14} - 6810 q^{15} - 327686 q^{16} + 97644 q^{17} - 784608 q^{18} + 473602 q^{19} + 1422066 q^{20} + 65592 q^{21} - 809556 q^{22} - 711654 q^{23} - 2383062 q^{24} + 1910964 q^{26} + 3751188 q^{27} + 4005248 q^{28} - 2657418 q^{29} - 6200118 q^{30} - 99196 q^{31} + 1902216 q^{32} + 999108 q^{33} + 4060674 q^{34} - 1397796 q^{35} - 11372454 q^{36} - 8096248 q^{37} + 13660674 q^{39} + 2352828 q^{40} - 1145094 q^{41} + 14113128 q^{42} + 18199452 q^{43} + 1762716 q^{44} - 3863376 q^{45} - 23702154 q^{46} - 21423408 q^{47} - 30089718 q^{48} - 18234288 q^{49} + 30161304 q^{50} - 40945708 q^{52} + 24470232 q^{53} + 80265120 q^{54} + 74753604 q^{55} + 84926940 q^{56} + 22501356 q^{57} - 46153494 q^{58} - 56069124 q^{59} - 225562428 q^{60} - 90554442 q^{61} + 2935614 q^{62} - 59182548 q^{63} + 54879114 q^{65} + 158753424 q^{66} + 31422418 q^{67} + 351553224 q^{68} + 163279656 q^{69} + 90147084 q^{70} - 46423914 q^{71} - 348800880 q^{72} - 175935974 q^{73} - 390463386 q^{74} - 364524840 q^{75} - 331167530 q^{76} + 464119008 q^{78} + 287989080 q^{79} + 897035604 q^{80} + 276106674 q^{81} + 272314014 q^{82} + 147421632 q^{83} - 118954092 q^{84} - 71261994 q^{85} - 611519460 q^{86} - 955994634 q^{87} - 1180586364 q^{88} - 422276046 q^{89} + 484120936 q^{91} + 1153271580 q^{92} + 1306731180 q^{93} + 1241176758 q^{94} + 318589986 q^{95} - 304944120 q^{96} + 101151350 q^{97} - 699182022 q^{98} - 1484420112 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(13))\)

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
13.9.d \(\chi_{13}(5, \cdot)\) 13.9.d.a 18 2
13.9.f \(\chi_{13}(2, \cdot)\) 13.9.f.a 32 4