## Defining parameters

 Level: $$N$$ = $$140 = 2^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$21$$ Sturm bound: $$2304$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(140))$$.

Total New Old
Modular forms 696 322 374
Cusp forms 457 266 191
Eisenstein series 239 56 183

## Trace form

 $$266q - 2q^{2} + 6q^{3} - 6q^{4} - 5q^{5} - 24q^{6} + 6q^{7} - 26q^{8} - 18q^{9} + O(q^{10})$$ $$266q - 2q^{2} + 6q^{3} - 6q^{4} - 5q^{5} - 24q^{6} + 6q^{7} - 26q^{8} - 18q^{9} - 30q^{10} - 6q^{11} - 36q^{12} - 32q^{13} - 30q^{14} - 10q^{15} - 26q^{16} - 6q^{17} - 6q^{18} + 6q^{19} - 4q^{20} - 54q^{21} + 12q^{22} - 6q^{23} + 12q^{24} - 57q^{25} - 8q^{26} - 24q^{27} + 42q^{28} - 84q^{29} - 6q^{30} - 30q^{31} - 22q^{32} - 102q^{33} - 49q^{35} - 90q^{36} - 50q^{37} - 48q^{38} - 48q^{39} - 16q^{40} - 76q^{41} - 36q^{42} + 12q^{43} - 48q^{44} - 76q^{45} - 48q^{46} - 18q^{47} - 12q^{48} + 18q^{49} - 14q^{50} - 18q^{51} + 44q^{52} - 6q^{53} + 96q^{54} + 42q^{55} + 6q^{56} + 60q^{57} + 80q^{58} + 42q^{59} + 120q^{60} + 6q^{61} + 96q^{62} + 114q^{63} + 102q^{64} + 42q^{65} + 132q^{66} + 78q^{67} + 144q^{68} + 108q^{69} + 162q^{70} + 72q^{71} + 222q^{72} + 50q^{73} + 144q^{74} + 99q^{75} + 120q^{76} + 6q^{77} + 72q^{78} + 54q^{79} + 88q^{80} + 12q^{81} + 52q^{82} - 12q^{83} + 84q^{84} - 6q^{85} + 60q^{86} + 48q^{88} - 150q^{89} + 48q^{90} - 36q^{91} - 12q^{92} - 150q^{93} + 48q^{94} - 107q^{95} + 24q^{96} - 112q^{97} - 10q^{98} - 120q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(140))$$

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
140.2.a $$\chi_{140}(1, \cdot)$$ 140.2.a.a 1 1
140.2.a.b 1
140.2.c $$\chi_{140}(139, \cdot)$$ 140.2.c.a 4 1
140.2.c.b 16
140.2.e $$\chi_{140}(29, \cdot)$$ 140.2.e.a 2 1
140.2.e.b 2
140.2.g $$\chi_{140}(111, \cdot)$$ 140.2.g.a 4 1
140.2.g.b 4
140.2.g.c 8
140.2.i $$\chi_{140}(81, \cdot)$$ 140.2.i.a 2 2
140.2.i.b 2
140.2.k $$\chi_{140}(43, \cdot)$$ 140.2.k.a 36 2
140.2.m $$\chi_{140}(13, \cdot)$$ 140.2.m.a 8 2
140.2.o $$\chi_{140}(31, \cdot)$$ 140.2.o.a 32 2
140.2.q $$\chi_{140}(9, \cdot)$$ 140.2.q.a 4 2
140.2.q.b 4
140.2.s $$\chi_{140}(19, \cdot)$$ 140.2.s.a 8 2
140.2.s.b 32
140.2.u $$\chi_{140}(17, \cdot)$$ 140.2.u.a 16 4
140.2.w $$\chi_{140}(23, \cdot)$$ 140.2.w.a 8 4
140.2.w.b 72

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(140))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(140)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$$$^{\oplus 2}$$