## Defining parameters

 Level: $$N$$ = $$169 = 13^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$14$$ Sturm bound: $$7098$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(169))$$.

Total New Old
Modular forms 2480 2440 40
Cusp forms 2252 2236 16
Eisenstein series 228 204 24

## Trace form

 $$2236 q - 66 q^{2} - 66 q^{3} - 66 q^{4} - 66 q^{5} - 66 q^{6} - 86 q^{7} - 138 q^{8} - 114 q^{9} + O(q^{10})$$ $$2236 q - 66 q^{2} - 66 q^{3} - 66 q^{4} - 66 q^{5} - 66 q^{6} - 86 q^{7} - 138 q^{8} - 114 q^{9} - 126 q^{10} - 78 q^{11} - 78 q^{12} - 60 q^{13} - 78 q^{14} + 6 q^{15} + 94 q^{16} - 54 q^{17} - 138 q^{18} - 98 q^{19} - 162 q^{20} - 222 q^{21} - 246 q^{22} - 114 q^{23} - 210 q^{24} - 78 q^{25} - 42 q^{26} + 18 q^{27} + 2 q^{28} - 162 q^{29} + 30 q^{30} - 118 q^{31} - 78 q^{32} + 138 q^{33} + 174 q^{34} - 6 q^{35} + 30 q^{36} - 22 q^{37} - 78 q^{38} - 144 q^{39} - 462 q^{40} - 342 q^{41} - 318 q^{42} - 438 q^{43} + 90 q^{44} - 186 q^{45} - 282 q^{46} + 66 q^{47} + 174 q^{48} + 66 q^{49} + 270 q^{50} - 78 q^{51} - 158 q^{52} - 246 q^{53} - 414 q^{54} - 6 q^{55} + 90 q^{56} + 42 q^{57} + 282 q^{58} + 138 q^{59} + 138 q^{60} + 762 q^{61} - 90 q^{62} - 54 q^{63} - 78 q^{64} - 186 q^{65} - 534 q^{66} - 146 q^{67} - 1110 q^{68} - 222 q^{69} - 342 q^{70} - 474 q^{71} - 126 q^{72} - 194 q^{73} - 414 q^{74} - 174 q^{75} - 266 q^{76} - 78 q^{77} + 258 q^{78} - 198 q^{79} + 378 q^{80} + 366 q^{81} - 126 q^{82} + 402 q^{83} + 282 q^{84} + 18 q^{85} + 330 q^{86} - 474 q^{87} + 330 q^{88} - 258 q^{89} - 78 q^{90} - 286 q^{91} + 1074 q^{92} - 198 q^{93} + 774 q^{94} + 78 q^{95} - 798 q^{96} - 298 q^{97} - 498 q^{98} - 798 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(169))$$

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
169.3.d $$\chi_{169}(70, \cdot)$$ 169.3.d.a 4 2
169.3.d.b 4
169.3.d.c 4
169.3.d.d 4
169.3.d.e 24
169.3.f $$\chi_{169}(19, \cdot)$$ 169.3.f.a 4 4
169.3.f.b 4
169.3.f.c 4
169.3.f.d 8
169.3.f.e 8
169.3.f.f 8
169.3.f.g 48
169.3.j $$\chi_{169}(5, \cdot)$$ 169.3.j.a 720 24
169.3.l $$\chi_{169}(2, \cdot)$$ 169.3.l.a 1392 48

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(169))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(169)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$