## Defining parameters

 Level: $$N$$ = $$169 = 13^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$8$$ Newform subspaces: $$14$$ Sturm bound: $$4732$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(169))$$.

Total New Old
Modular forms 1297 1271 26
Cusp forms 1070 1066 4
Eisenstein series 227 205 22

## Trace form

 $$1066q - 69q^{2} - 70q^{3} - 73q^{4} - 72q^{5} - 78q^{6} - 70q^{7} - 63q^{8} - 63q^{9} + O(q^{10})$$ $$1066q - 69q^{2} - 70q^{3} - 73q^{4} - 72q^{5} - 78q^{6} - 70q^{7} - 63q^{8} - 63q^{9} - 54q^{10} - 66q^{11} - 42q^{12} - 60q^{13} - 126q^{14} - 66q^{15} - 57q^{16} - 66q^{17} - 39q^{18} - 46q^{19} - 42q^{20} - 46q^{21} - 42q^{22} - 66q^{23} - 6q^{24} - 55q^{25} - 42q^{26} - 94q^{27} - 22q^{28} - 42q^{29} + 6q^{30} - 46q^{31} + 3q^{32} - 30q^{33} - 24q^{34} - 30q^{35} + 11q^{36} - 70q^{37} - 42q^{38} - 40q^{39} - 72q^{40} - 18q^{41} + 18q^{42} - 18q^{43} + 6q^{44} + 6q^{45} + 30q^{46} - 30q^{47} + 26q^{48} - 7q^{49} + 27q^{50} + 18q^{51} + 11q^{52} - 84q^{53} + 18q^{54} - 6q^{55} + 42q^{56} + 2q^{57} - 6q^{58} - 42q^{59} + 90q^{60} - 14q^{61} + 6q^{62} + 26q^{63} + 53q^{64} + 9q^{65} - 6q^{66} - 22q^{67} + 54q^{68} - 6q^{69} + 66q^{70} - 18q^{71} + 111q^{72} - 4q^{73} + 66q^{74} + 54q^{75} + 74q^{76} + 18q^{77} + 102q^{78} - 86q^{79} + 138q^{80} + 21q^{81} + 30q^{82} + 6q^{83} + 146q^{84} + 12q^{85} + 54q^{86} + 54q^{87} + 102q^{88} + 36q^{89} + 144q^{90} + 26q^{91} - 6q^{92} + 74q^{93} + 78q^{94} + 30q^{95} + 174q^{96} - 4q^{97} + 51q^{98} + 78q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(169))$$

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
169.2.a $$\chi_{169}(1, \cdot)$$ 169.2.a.a 2 1
169.2.a.b 3
169.2.a.c 3
169.2.b $$\chi_{169}(168, \cdot)$$ 169.2.b.a 2 1
169.2.b.b 6
169.2.c $$\chi_{169}(22, \cdot)$$ 169.2.c.a 4 2
169.2.c.b 6
169.2.c.c 6
169.2.e $$\chi_{169}(23, \cdot)$$ 169.2.e.a 2 2
169.2.e.b 12
169.2.g $$\chi_{169}(14, \cdot)$$ 169.2.g.a 156 12
169.2.h $$\chi_{169}(12, \cdot)$$ 169.2.h.a 168 12
169.2.i $$\chi_{169}(3, \cdot)$$ 169.2.i.a 336 24
169.2.k $$\chi_{169}(4, \cdot)$$ 169.2.k.a 360 24

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(169))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(169)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$