## Defining parameters

 Level: $$N$$ = $$35 = 5 \cdot 7$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$480$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(35))$$.

Total New Old
Modular forms 216 182 34
Cusp forms 168 154 14
Eisenstein series 48 28 20

## Trace form

 $$154 q - 2 q^{2} + 58 q^{4} - 62 q^{5} - 72 q^{6} - 58 q^{7} - 450 q^{8} - 126 q^{9} + O(q^{10})$$ $$154 q - 2 q^{2} + 58 q^{4} - 62 q^{5} - 72 q^{6} - 58 q^{7} - 450 q^{8} - 126 q^{9} + 212 q^{10} + 536 q^{11} + 828 q^{12} - 568 q^{13} - 1230 q^{14} + 348 q^{15} + 358 q^{16} + 484 q^{17} - 642 q^{18} - 1296 q^{19} - 852 q^{20} + 1452 q^{21} + 2884 q^{22} + 1540 q^{23} + 1620 q^{24} + 3208 q^{25} + 1376 q^{26} + 888 q^{27} - 3518 q^{28} - 2100 q^{29} - 13572 q^{30} - 13364 q^{31} - 15806 q^{32} - 8244 q^{33} + 5210 q^{35} + 18198 q^{36} + 11140 q^{37} + 23124 q^{38} + 18744 q^{39} + 24948 q^{40} + 12272 q^{41} + 36 q^{42} - 9116 q^{43} - 16776 q^{44} - 13692 q^{45} - 19628 q^{46} - 18524 q^{47} - 19500 q^{48} - 2854 q^{49} - 15542 q^{50} + 9396 q^{51} + 17228 q^{52} + 33496 q^{53} + 37572 q^{54} + 29524 q^{55} + 37926 q^{56} - 528 q^{57} + 6528 q^{58} - 13704 q^{59} - 43056 q^{60} - 39236 q^{61} - 54424 q^{62} - 59586 q^{63} - 57962 q^{64} - 58100 q^{65} - 47664 q^{66} - 35724 q^{67} - 58408 q^{68} + 23208 q^{70} + 52460 q^{71} + 81390 q^{72} + 42644 q^{73} + 75684 q^{74} + 97638 q^{75} + 93720 q^{76} + 89380 q^{77} + 108648 q^{78} + 79180 q^{79} + 65656 q^{80} - 39210 q^{81} - 6500 q^{82} - 42788 q^{83} - 67716 q^{84} - 50560 q^{85} - 43228 q^{86} - 17904 q^{87} + 35088 q^{88} - 25692 q^{89} + 4680 q^{90} - 28960 q^{91} - 59996 q^{92} - 61236 q^{93} - 140028 q^{94} - 89946 q^{95} - 141144 q^{96} - 46888 q^{97} - 60498 q^{98} - 70428 q^{99} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(35))$$

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
35.5.c $$\chi_{35}(34, \cdot)$$ 35.5.c.a 1 1
35.5.c.b 1
35.5.c.c 2
35.5.c.d 2
35.5.c.e 8
35.5.d $$\chi_{35}(6, \cdot)$$ 35.5.d.a 12 1
35.5.g $$\chi_{35}(8, \cdot)$$ 35.5.g.a 24 2
35.5.h $$\chi_{35}(26, \cdot)$$ 35.5.h.a 20 2
35.5.i $$\chi_{35}(19, \cdot)$$ 35.5.i.a 28 2
35.5.l $$\chi_{35}(2, \cdot)$$ 35.5.l.a 56 4

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(35))$$ into lower level spaces

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