## 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

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

## 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

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