## Defining parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ 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

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

## 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