## Defining parameters

 Level: $$N$$ = $$13$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$2$$ Newforms: $$2$$ Sturm bound: $$98$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(13))$$.

Total New Old
Modular forms 48 48 0
Cusp forms 36 36 0
Eisenstein series 12 12 0

## Trace form

 $$36q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} + 714q^{7} - 2310q^{8} - 6q^{9} + O(q^{10})$$ $$36q - 6q^{2} - 6q^{3} - 6q^{4} - 6q^{5} - 6q^{6} + 714q^{7} - 2310q^{8} - 6q^{9} + 4674q^{10} + 1914q^{11} - 5478q^{13} - 9132q^{14} - 7782q^{15} - 6q^{16} + 20898q^{17} + 9624q^{18} - 12270q^{19} - 33678q^{20} + 3114q^{21} + 66684q^{22} + 29634q^{23} + 78762q^{24} - 57756q^{26} - 127524q^{27} - 36480q^{28} - 89082q^{29} - 247734q^{30} - 57102q^{31} + 115944q^{32} + 261594q^{33} + 361794q^{34} + 263634q^{35} + 105066q^{36} - 71682q^{37} + 166362q^{39} - 379524q^{40} - 547632q^{41} - 984840q^{42} - 480486q^{43} - 308340q^{44} + 312804q^{45} + 967638q^{46} + 702474q^{47} + 1606410q^{48} + 854562q^{49} + 839136q^{50} - 1350204q^{52} - 887940q^{53} - 2142480q^{54} - 1469526q^{55} - 1461108q^{56} - 543486q^{57} + 113826q^{58} + 573594q^{59} + 3294612q^{60} + 1782372q^{61} + 3741966q^{62} + 2202042q^{63} - 2077032q^{65} - 5495280q^{66} - 2932278q^{67} - 2882184q^{68} - 1475766q^{69} + 332412q^{70} + 1481082q^{71} + 3080544q^{72} + 2914554q^{73} + 5533470q^{74} + 3466008q^{75} + 1769142q^{76} - 1496064q^{78} - 1809660q^{79} - 7899708q^{80} - 4789212q^{81} - 3540426q^{82} - 2747166q^{83} - 268140q^{84} + 3615312q^{85} + 3572508q^{86} + 2640762q^{87} + 4478052q^{88} - 344622q^{89} - 1858038q^{91} + 2026812q^{92} - 511782q^{93} - 4686090q^{94} - 2753718q^{95} - 3082824q^{96} + 3241602q^{97} + 6023610q^{98} + 7231194q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(13))$$

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
13.7.d $$\chi_{13}(5, \cdot)$$ 13.7.d.a 12 2
13.7.f $$\chi_{13}(2, \cdot)$$ 13.7.f.a 24 4