## Defining parameters

 Level: $$N$$ = $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$10$$ Newform subspaces: $$24$$ Sturm bound: $$6144$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(160))$$.

Total New Old
Modular forms 2432 1218 1214
Cusp forms 2176 1158 1018
Eisenstein series 256 60 196

## Trace form

 $$1158 q - 8 q^{2} - 4 q^{3} - 8 q^{4} - 14 q^{5} - 24 q^{6} - 36 q^{7} - 8 q^{8} - 106 q^{9} + O(q^{10})$$ $$1158 q - 8 q^{2} - 4 q^{3} - 8 q^{4} - 14 q^{5} - 24 q^{6} - 36 q^{7} - 8 q^{8} - 106 q^{9} - 132 q^{10} - 16 q^{11} + 88 q^{12} + 228 q^{13} + 408 q^{14} + 100 q^{15} + 576 q^{16} + 356 q^{17} + 352 q^{18} - 8 q^{19} - 92 q^{20} - 536 q^{21} - 400 q^{22} - 1268 q^{23} + 80 q^{24} - 230 q^{25} - 64 q^{26} + 416 q^{27} - 768 q^{28} - 180 q^{29} - 1204 q^{30} + 1848 q^{31} - 1248 q^{32} + 824 q^{33} - 1072 q^{34} + 904 q^{35} - 2944 q^{36} - 356 q^{37} - 1976 q^{38} - 1072 q^{39} + 520 q^{40} + 244 q^{41} + 4512 q^{42} - 756 q^{43} + 4072 q^{44} + 1574 q^{45} + 2856 q^{46} + 924 q^{47} + 4880 q^{48} + 510 q^{49} + 3548 q^{50} + 1256 q^{51} + 5032 q^{52} - 2052 q^{53} + 2160 q^{54} - 556 q^{55} - 2416 q^{56} - 1864 q^{57} - 6416 q^{58} - 2760 q^{59} - 4744 q^{60} + 1844 q^{61} - 6784 q^{62} - 3740 q^{63} - 2192 q^{64} + 2436 q^{65} + 1160 q^{66} - 2236 q^{67} + 384 q^{68} + 3544 q^{69} + 168 q^{70} - 976 q^{71} - 2120 q^{72} - 884 q^{73} - 1688 q^{74} - 1104 q^{75} + 1640 q^{76} - 8264 q^{77} + 408 q^{78} - 5008 q^{79} - 2344 q^{80} - 6874 q^{81} + 1112 q^{82} - 7564 q^{83} - 4832 q^{84} - 3748 q^{85} - 2912 q^{86} + 4536 q^{87} - 3744 q^{88} + 4268 q^{89} + 24 q^{90} + 8880 q^{91} - 9232 q^{92} - 144 q^{93} - 11024 q^{94} + 8656 q^{95} - 10912 q^{96} + 2516 q^{97} - 9840 q^{98} + 10832 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(160))$$

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
160.4.a $$\chi_{160}(1, \cdot)$$ 160.4.a.a 1 1
160.4.a.b 1
160.4.a.c 2
160.4.a.d 2
160.4.a.e 2
160.4.a.f 2
160.4.a.g 2
160.4.c $$\chi_{160}(129, \cdot)$$ 160.4.c.a 2 1
160.4.c.b 4
160.4.c.c 4
160.4.c.d 8
160.4.d $$\chi_{160}(81, \cdot)$$ 160.4.d.a 12 1
160.4.f $$\chi_{160}(49, \cdot)$$ 160.4.f.a 16 1
160.4.j $$\chi_{160}(87, \cdot)$$ None 0 2
160.4.l $$\chi_{160}(41, \cdot)$$ None 0 2
160.4.n $$\chi_{160}(63, \cdot)$$ 160.4.n.a 2 2
160.4.n.b 2
160.4.n.c 8
160.4.n.d 8
160.4.n.e 8
160.4.n.f 8
160.4.o $$\chi_{160}(47, \cdot)$$ 160.4.o.a 32 2
160.4.q $$\chi_{160}(9, \cdot)$$ None 0 2
160.4.s $$\chi_{160}(7, \cdot)$$ None 0 2
160.4.u $$\chi_{160}(43, \cdot)$$ 160.4.u.a 280 4
160.4.x $$\chi_{160}(21, \cdot)$$ 160.4.x.a 192 4
160.4.z $$\chi_{160}(29, \cdot)$$ 160.4.z.a 280 4
160.4.ba $$\chi_{160}(3, \cdot)$$ 160.4.ba.a 280 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(160))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(160)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(40))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 2}$$