## Defining parameters

 Level: $$N$$ = $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$16$$ Sturm bound: $$1152$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(70))$$.

Total New Old
Modular forms 480 126 354
Cusp forms 384 126 258
Eisenstein series 96 0 96

## Trace form

 $$126 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 34 q^{5} + 88 q^{6} + 100 q^{7} - 16 q^{8} - 250 q^{9} + O(q^{10})$$ $$126 q - 4 q^{2} - 8 q^{3} + 8 q^{4} + 34 q^{5} + 88 q^{6} + 100 q^{7} - 16 q^{8} - 250 q^{9} - 136 q^{10} + 4 q^{11} - 32 q^{12} - 16 q^{13} - 152 q^{14} + 172 q^{15} - 96 q^{16} + 168 q^{17} + 188 q^{18} + 80 q^{19} + 8 q^{21} + 384 q^{22} + 324 q^{23} + 288 q^{24} + 1384 q^{25} + 352 q^{26} + 592 q^{27} - 32 q^{28} - 804 q^{29} - 936 q^{30} - 1268 q^{31} - 64 q^{32} - 2508 q^{33} - 1592 q^{34} - 2528 q^{35} - 1128 q^{36} - 2032 q^{37} - 872 q^{38} - 2008 q^{39} + 96 q^{40} + 1876 q^{41} + 872 q^{42} + 2792 q^{43} + 464 q^{44} + 5640 q^{45} + 2624 q^{46} + 3372 q^{47} + 448 q^{48} + 830 q^{49} + 676 q^{50} - 700 q^{51} - 160 q^{52} - 3048 q^{53} - 192 q^{54} + 1372 q^{55} - 480 q^{56} + 1952 q^{57} + 1368 q^{58} + 1576 q^{59} + 504 q^{60} - 5436 q^{61} - 848 q^{62} - 2348 q^{63} - 640 q^{64} - 452 q^{65} + 1600 q^{66} + 116 q^{67} + 672 q^{68} - 2312 q^{69} + 276 q^{70} + 2016 q^{71} + 752 q^{72} - 5008 q^{73} - 1000 q^{74} - 2878 q^{75} - 1168 q^{76} - 5340 q^{77} - 4160 q^{78} - 2644 q^{79} + 544 q^{80} + 3402 q^{81} + 3720 q^{82} + 5292 q^{83} + 2112 q^{84} + 9868 q^{85} + 1008 q^{86} + 14184 q^{87} + 1152 q^{88} + 8120 q^{89} + 5208 q^{90} + 4820 q^{91} + 2304 q^{92} + 6548 q^{93} + 4832 q^{94} + 1570 q^{95} - 128 q^{96} - 676 q^{97} - 2308 q^{98} - 5056 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(70))$$

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
70.4.a $$\chi_{70}(1, \cdot)$$ 70.4.a.a 1 1
70.4.a.b 1
70.4.a.c 1
70.4.a.d 1
70.4.a.e 1
70.4.a.f 1
70.4.c $$\chi_{70}(29, \cdot)$$ 70.4.c.a 2 1
70.4.c.b 6
70.4.e $$\chi_{70}(11, \cdot)$$ 70.4.e.a 2 2
70.4.e.b 2
70.4.e.c 2
70.4.e.d 4
70.4.e.e 6
70.4.g $$\chi_{70}(13, \cdot)$$ 70.4.g.a 24 2
70.4.i $$\chi_{70}(9, \cdot)$$ 70.4.i.a 24 2
70.4.k $$\chi_{70}(3, \cdot)$$ 70.4.k.a 48 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(70))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(70)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$