Properties

Label 13.4
Level 13
Weight 4
Dimension 15
Nonzero newspaces 4
Newforms 7
Sturm bound 56
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 4 \)
Newforms: \( 7 \)
Sturm bound: \(56\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(13))\).

Total New Old
Modular forms 27 25 2
Cusp forms 15 15 0
Eisenstein series 12 10 2

Trace form

\(15q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 42q^{7} \) \(\mathstrut -\mathstrut 78q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(15q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 6q^{6} \) \(\mathstrut -\mathstrut 42q^{7} \) \(\mathstrut -\mathstrut 78q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut 84q^{10} \) \(\mathstrut +\mathstrut 54q^{11} \) \(\mathstrut +\mathstrut 228q^{12} \) \(\mathstrut +\mathstrut 138q^{13} \) \(\mathstrut +\mathstrut 108q^{14} \) \(\mathstrut +\mathstrut 66q^{15} \) \(\mathstrut -\mathstrut 6q^{16} \) \(\mathstrut -\mathstrut 195q^{17} \) \(\mathstrut -\mathstrut 834q^{18} \) \(\mathstrut -\mathstrut 450q^{19} \) \(\mathstrut -\mathstrut 522q^{20} \) \(\mathstrut -\mathstrut 6q^{21} \) \(\mathstrut +\mathstrut 384q^{22} \) \(\mathstrut +\mathstrut 222q^{23} \) \(\mathstrut +\mathstrut 1050q^{24} \) \(\mathstrut +\mathstrut 363q^{25} \) \(\mathstrut +\mathstrut 744q^{26} \) \(\mathstrut +\mathstrut 816q^{27} \) \(\mathstrut -\mathstrut 60q^{28} \) \(\mathstrut -\mathstrut 435q^{29} \) \(\mathstrut -\mathstrut 582q^{30} \) \(\mathstrut -\mathstrut 318q^{31} \) \(\mathstrut -\mathstrut 660q^{32} \) \(\mathstrut -\mathstrut 606q^{33} \) \(\mathstrut -\mathstrut 702q^{34} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut 162q^{36} \) \(\mathstrut +\mathstrut 177q^{37} \) \(\mathstrut +\mathstrut 1356q^{38} \) \(\mathstrut +\mathstrut 66q^{39} \) \(\mathstrut -\mathstrut 228q^{40} \) \(\mathstrut -\mathstrut 477q^{41} \) \(\mathstrut -\mathstrut 636q^{42} \) \(\mathstrut -\mathstrut 426q^{43} \) \(\mathstrut +\mathstrut 228q^{44} \) \(\mathstrut +\mathstrut 279q^{45} \) \(\mathstrut +\mathstrut 822q^{46} \) \(\mathstrut +\mathstrut 126q^{47} \) \(\mathstrut -\mathstrut 390q^{48} \) \(\mathstrut +\mathstrut 966q^{49} \) \(\mathstrut -\mathstrut 102q^{50} \) \(\mathstrut -\mathstrut 396q^{51} \) \(\mathstrut -\mathstrut 336q^{52} \) \(\mathstrut +\mathstrut 936q^{53} \) \(\mathstrut +\mathstrut 720q^{54} \) \(\mathstrut +\mathstrut 1398q^{55} \) \(\mathstrut +\mathstrut 996q^{56} \) \(\mathstrut +\mathstrut 1158q^{57} \) \(\mathstrut +\mathstrut 1020q^{58} \) \(\mathstrut +\mathstrut 486q^{59} \) \(\mathstrut +\mathstrut 1152q^{60} \) \(\mathstrut -\mathstrut 1137q^{61} \) \(\mathstrut -\mathstrut 2622q^{62} \) \(\mathstrut -\mathstrut 3186q^{63} \) \(\mathstrut -\mathstrut 4692q^{64} \) \(\mathstrut -\mathstrut 4065q^{65} \) \(\mathstrut -\mathstrut 4152q^{66} \) \(\mathstrut -\mathstrut 918q^{67} \) \(\mathstrut +\mathstrut 1320q^{68} \) \(\mathstrut +\mathstrut 2022q^{69} \) \(\mathstrut +\mathstrut 1980q^{70} \) \(\mathstrut +\mathstrut 1938q^{71} \) \(\mathstrut +\mathstrut 2940q^{72} \) \(\mathstrut +\mathstrut 2154q^{73} \) \(\mathstrut +\mathstrut 2568q^{74} \) \(\mathstrut +\mathstrut 612q^{75} \) \(\mathstrut -\mathstrut 1842q^{76} \) \(\mathstrut +\mathstrut 372q^{77} \) \(\mathstrut +\mathstrut 2940q^{78} \) \(\mathstrut -\mathstrut 972q^{79} \) \(\mathstrut +\mathstrut 3384q^{80} \) \(\mathstrut +\mathstrut 456q^{81} \) \(\mathstrut +\mathstrut 3552q^{82} \) \(\mathstrut +\mathstrut 3426q^{83} \) \(\mathstrut +\mathstrut 2268q^{84} \) \(\mathstrut +\mathstrut 1011q^{85} \) \(\mathstrut -\mathstrut 2808q^{86} \) \(\mathstrut -\mathstrut 3906q^{87} \) \(\mathstrut -\mathstrut 444q^{88} \) \(\mathstrut -\mathstrut 3726q^{89} \) \(\mathstrut -\mathstrut 6936q^{90} \) \(\mathstrut -\mathstrut 2730q^{91} \) \(\mathstrut -\mathstrut 4812q^{92} \) \(\mathstrut -\mathstrut 4194q^{93} \) \(\mathstrut -\mathstrut 1722q^{94} \) \(\mathstrut -\mathstrut 1962q^{95} \) \(\mathstrut +\mathstrut 1548q^{96} \) \(\mathstrut +\mathstrut 1806q^{97} \) \(\mathstrut -\mathstrut 2730q^{98} \) \(\mathstrut +\mathstrut 5190q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(13))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
13.4.a \(\chi_{13}(1, \cdot)\) 13.4.a.a 1 1
13.4.a.b 2
13.4.b \(\chi_{13}(12, \cdot)\) 13.4.b.a 2 1
13.4.c \(\chi_{13}(3, \cdot)\) 13.4.c.a 2 2
13.4.c.b 4
13.4.e \(\chi_{13}(4, \cdot)\) 13.4.e.a 2 2
13.4.e.b 2