## Defining parameters

 Level: $$N$$ = $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$32$$ Sturm bound: $$2688$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(156))$$.

Total New Old
Modular forms 792 310 482
Cusp forms 553 270 283
Eisenstein series 239 40 199

## Trace form

 $$270 q - 12 q^{4} - 6 q^{6} + 4 q^{7} - 10 q^{9} + O(q^{10})$$ $$270 q - 12 q^{4} - 6 q^{6} + 4 q^{7} - 10 q^{9} - 12 q^{10} + 12 q^{11} - 12 q^{12} + 12 q^{15} - 12 q^{16} + 6 q^{17} - 24 q^{18} - 8 q^{19} - 48 q^{20} - 34 q^{21} - 72 q^{22} - 24 q^{23} - 54 q^{24} - 78 q^{25} - 60 q^{26} - 36 q^{27} - 72 q^{28} - 30 q^{29} - 54 q^{30} - 24 q^{31} - 60 q^{32} - 54 q^{33} - 60 q^{34} - 24 q^{35} - 24 q^{36} - 30 q^{37} - 28 q^{39} + 24 q^{40} - 66 q^{41} + 12 q^{42} - 32 q^{43} + 60 q^{44} - 60 q^{45} + 108 q^{46} - 24 q^{47} + 42 q^{48} - 176 q^{49} + 108 q^{50} - 24 q^{51} + 96 q^{52} - 48 q^{53} - 18 q^{54} - 12 q^{55} + 108 q^{56} - 58 q^{57} + 84 q^{58} + 12 q^{59} + 84 q^{60} - 90 q^{61} + 60 q^{62} + 62 q^{63} + 48 q^{64} - 18 q^{65} + 72 q^{66} + 40 q^{67} + 66 q^{69} + 48 q^{70} + 24 q^{71} + 72 q^{72} + 16 q^{73} + 104 q^{75} - 12 q^{76} + 72 q^{77} + 108 q^{78} + 72 q^{79} + 50 q^{81} - 12 q^{82} + 36 q^{83} + 120 q^{84} + 54 q^{85} + 48 q^{86} + 84 q^{87} + 48 q^{88} + 72 q^{89} + 144 q^{90} + 52 q^{91} + 72 q^{92} + 74 q^{93} + 84 q^{94} + 48 q^{95} + 120 q^{96} + 60 q^{97} + 108 q^{98} - 18 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(156))$$

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
156.2.a $$\chi_{156}(1, \cdot)$$ 156.2.a.a 1 1
156.2.a.b 1
156.2.b $$\chi_{156}(25, \cdot)$$ 156.2.b.a 2 1
156.2.b.b 2
156.2.c $$\chi_{156}(131, \cdot)$$ 156.2.c.a 2 1
156.2.c.b 2
156.2.c.c 8
156.2.c.d 12
156.2.h $$\chi_{156}(155, \cdot)$$ 156.2.h.a 8 1
156.2.h.b 16
156.2.i $$\chi_{156}(61, \cdot)$$ 156.2.i.a 2 2
156.2.k $$\chi_{156}(31, \cdot)$$ 156.2.k.a 2 2
156.2.k.b 2
156.2.k.c 2
156.2.k.d 2
156.2.k.e 10
156.2.k.f 10
156.2.m $$\chi_{156}(5, \cdot)$$ 156.2.m.a 4 2
156.2.m.b 4
156.2.p $$\chi_{156}(35, \cdot)$$ 156.2.p.a 8 2
156.2.p.b 40
156.2.q $$\chi_{156}(49, \cdot)$$ 156.2.q.a 2 2
156.2.q.b 4
156.2.r $$\chi_{156}(23, \cdot)$$ 156.2.r.a 4 2
156.2.r.b 4
156.2.r.c 40
156.2.u $$\chi_{156}(41, \cdot)$$ 156.2.u.a 4 4
156.2.u.b 16
156.2.w $$\chi_{156}(7, \cdot)$$ 156.2.w.a 4 4
156.2.w.b 4
156.2.w.c 24
156.2.w.d 24

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(156))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(156)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 2}$$