## Defining parameters

 Level: $$N$$ = $$136 = 2^{3} \cdot 17$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$9$$ Newform subspaces: $$18$$ Sturm bound: $$4608$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(136))$$.

Total New Old
Modular forms 1824 994 830
Cusp forms 1632 934 698
Eisenstein series 192 60 132

## Trace form

 $$934 q - 12 q^{2} - 8 q^{3} + 8 q^{4} + 4 q^{5} - 72 q^{6} - 32 q^{7} - 96 q^{8} - 6 q^{9} + O(q^{10})$$ $$934 q - 12 q^{2} - 8 q^{3} + 8 q^{4} + 4 q^{5} - 72 q^{6} - 32 q^{7} - 96 q^{8} - 6 q^{9} + 96 q^{10} + 72 q^{11} + 96 q^{12} - 44 q^{13} - 48 q^{14} - 256 q^{15} - 48 q^{16} - 54 q^{17} - 36 q^{18} - 104 q^{19} - 240 q^{20} + 192 q^{21} + 152 q^{22} + 704 q^{23} + 208 q^{24} - 622 q^{25} - 576 q^{26} - 536 q^{27} - 208 q^{28} - 136 q^{29} + 208 q^{30} + 272 q^{31} + 688 q^{32} + 336 q^{33} - 44 q^{34} + 1056 q^{35} - 40 q^{36} + 996 q^{37} + 376 q^{38} + 1376 q^{39} - 464 q^{40} + 184 q^{41} + 432 q^{42} - 1488 q^{43} - 352 q^{44} - 2024 q^{45} - 624 q^{46} - 2416 q^{47} - 928 q^{48} + 618 q^{49} + 36 q^{50} + 184 q^{51} + 1088 q^{52} + 2200 q^{53} + 7728 q^{54} + 4752 q^{55} + 5824 q^{56} + 568 q^{57} + 4240 q^{58} + 1560 q^{59} - 960 q^{60} - 2300 q^{61} - 3264 q^{62} - 6288 q^{63} - 6448 q^{64} - 2916 q^{65} - 13200 q^{66} - 3592 q^{67} - 11976 q^{68} - 6688 q^{69} - 11648 q^{70} - 3824 q^{71} - 9712 q^{72} + 392 q^{73} - 4992 q^{74} - 2568 q^{75} - 240 q^{76} + 3216 q^{77} + 7808 q^{78} + 6272 q^{79} + 12640 q^{80} + 6370 q^{81} + 11144 q^{82} + 5888 q^{83} + 15488 q^{84} - 2504 q^{85} + 4616 q^{86} - 7024 q^{87} - 688 q^{88} - 5164 q^{89} + 1872 q^{90} - 1648 q^{91} - 3664 q^{92} + 576 q^{93} + 1328 q^{94} + 6448 q^{95} + 880 q^{96} - 460 q^{97} - 1132 q^{98} + 6288 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(136))$$

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
136.4.a $$\chi_{136}(1, \cdot)$$ 136.4.a.a 2 1
136.4.a.b 3
136.4.a.c 3
136.4.a.d 4
136.4.b $$\chi_{136}(33, \cdot)$$ 136.4.b.a 6 1
136.4.b.b 8
136.4.c $$\chi_{136}(69, \cdot)$$ 136.4.c.a 24 1
136.4.c.b 24
136.4.h $$\chi_{136}(101, \cdot)$$ 136.4.h.a 52 1
136.4.i $$\chi_{136}(13, \cdot)$$ 136.4.i.a 104 2
136.4.k $$\chi_{136}(81, \cdot)$$ 136.4.k.a 14 2
136.4.k.b 14
136.4.n $$\chi_{136}(9, \cdot)$$ 136.4.n.a 24 4
136.4.n.b 28
136.4.o $$\chi_{136}(53, \cdot)$$ 136.4.o.a 208 4
136.4.r $$\chi_{136}(7, \cdot)$$ None 0 8
136.4.s $$\chi_{136}(3, \cdot)$$ 136.4.s.a 8 8
136.4.s.b 8
136.4.s.c 400

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(136))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(136)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(68))$$$$^{\oplus 2}$$