# Properties

 Label 169.4 Level 169 Weight 4 Dimension 3405 Nonzero newspaces 8 Newform subspaces 43 Sturm bound 9464 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$169 = 13^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$8$$ Newform subspaces: $$43$$ Sturm bound: $$9464$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(169))$$.

Total New Old
Modular forms 3663 3610 53
Cusp forms 3435 3405 30
Eisenstein series 228 205 23

## Trace form

 $$3405 q - 66 q^{2} - 66 q^{3} - 66 q^{4} - 66 q^{5} - 66 q^{6} + 6 q^{7} + 78 q^{8} - 66 q^{9} + O(q^{10})$$ $$3405 q - 66 q^{2} - 66 q^{3} - 66 q^{4} - 66 q^{5} - 66 q^{6} + 6 q^{7} + 78 q^{8} - 66 q^{9} - 246 q^{10} - 186 q^{11} - 534 q^{12} - 216 q^{13} - 366 q^{14} - 210 q^{15} - 66 q^{16} + 312 q^{17} + 1590 q^{18} + 822 q^{19} + 966 q^{20} - 66 q^{21} - 846 q^{22} - 522 q^{23} - 2178 q^{24} - 804 q^{25} - 822 q^{26} - 1782 q^{27} + 42 q^{28} + 792 q^{29} + 1086 q^{30} + 558 q^{31} + 1242 q^{32} + 1134 q^{33} + 1326 q^{34} - 66 q^{35} + 246 q^{36} - 432 q^{37} - 2790 q^{38} - 144 q^{39} + 306 q^{40} + 876 q^{41} + 1194 q^{42} + 774 q^{43} - 534 q^{44} - 636 q^{45} - 1722 q^{46} - 330 q^{47} + 702 q^{48} - 2010 q^{49} + 126 q^{50} + 714 q^{51} + 258 q^{52} - 2022 q^{53} - 1518 q^{54} - 2874 q^{55} - 2070 q^{56} - 2394 q^{57} - 2118 q^{58} - 1050 q^{59} - 2382 q^{60} + 2196 q^{61} + 5166 q^{62} + 6294 q^{63} + 9306 q^{64} + 3987 q^{65} + 8154 q^{66} + 1758 q^{67} - 2718 q^{68} - 4122 q^{69} - 4038 q^{70} - 3954 q^{71} - 5958 q^{72} - 4386 q^{73} - 5214 q^{74} - 1302 q^{75} + 3606 q^{76} - 822 q^{77} - 3018 q^{78} + 1794 q^{79} - 6846 q^{80} - 990 q^{81} - 7182 q^{82} - 6930 q^{83} - 4614 q^{84} - 2100 q^{85} + 5538 q^{86} + 7734 q^{87} + 810 q^{88} + 7374 q^{89} + 13794 q^{90} + 2652 q^{91} + 9474 q^{92} + 8310 q^{93} + 3366 q^{94} + 3846 q^{95} - 3174 q^{96} - 3690 q^{97} + 5382 q^{98} - 10458 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(169))$$

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
169.4.a $$\chi_{169}(1, \cdot)$$ 169.4.a.a 1 1
169.4.a.b 1
169.4.a.c 1
169.4.a.d 1
169.4.a.e 1
169.4.a.f 2
169.4.a.g 2
169.4.a.h 2
169.4.a.i 2
169.4.a.j 2
169.4.a.k 9
169.4.a.l 9
169.4.b $$\chi_{169}(168, \cdot)$$ 169.4.b.a 2 1
169.4.b.b 2
169.4.b.c 2
169.4.b.d 2
169.4.b.e 4
169.4.b.f 4
169.4.b.g 18
169.4.c $$\chi_{169}(22, \cdot)$$ 169.4.c.a 2 2
169.4.c.b 2
169.4.c.c 2
169.4.c.d 2
169.4.c.e 2
169.4.c.f 4
169.4.c.g 4
169.4.c.h 4
169.4.c.i 4
169.4.c.j 4
169.4.c.k 18
169.4.c.l 18
169.4.e $$\chi_{169}(23, \cdot)$$ 169.4.e.a 2 2
169.4.e.b 2
169.4.e.c 4
169.4.e.d 4
169.4.e.e 4
169.4.e.f 8
169.4.e.g 8
169.4.e.h 36
169.4.g $$\chi_{169}(14, \cdot)$$ 169.4.g.a 540 12
169.4.h $$\chi_{169}(12, \cdot)$$ 169.4.h.a 528 12
169.4.i $$\chi_{169}(3, \cdot)$$ 169.4.i.a 1080 24
169.4.k $$\chi_{169}(4, \cdot)$$ 169.4.k.a 1056 24

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(169))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(169)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$