## 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

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

## 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

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