## Defining parameters

 Level: $$N$$ = $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$27$$ Sturm bound: $$4032$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(156))$$.

Total New Old
Modular forms 1464 606 858
Cusp forms 1224 558 666
Eisenstein series 240 48 192

## Trace form

 $$558 q + 4 q^{2} + 6 q^{3} - 4 q^{4} + 8 q^{5} - 18 q^{6} - 24 q^{7} - 32 q^{8} - 24 q^{9} + O(q^{10})$$ $$558 q + 4 q^{2} + 6 q^{3} - 4 q^{4} + 8 q^{5} - 18 q^{6} - 24 q^{7} - 32 q^{8} - 24 q^{9} - 20 q^{10} - 12 q^{11} + 24 q^{12} + 18 q^{13} + 48 q^{14} + 36 q^{15} + 20 q^{16} + 68 q^{17} + 108 q^{18} + 204 q^{19} + 224 q^{20} + 66 q^{21} + 120 q^{22} + 48 q^{23} + 42 q^{24} + 34 q^{25} - 56 q^{26} - 18 q^{27} - 288 q^{28} - 16 q^{29} - 222 q^{30} - 52 q^{31} - 356 q^{32} - 66 q^{33} - 404 q^{34} - 240 q^{35} - 228 q^{36} - 252 q^{37} - 144 q^{38} - 42 q^{39} - 488 q^{40} + 428 q^{41} - 336 q^{42} + 300 q^{43} - 324 q^{44} - 138 q^{45} - 420 q^{46} + 96 q^{47} - 246 q^{48} + 366 q^{49} - 192 q^{50} - 312 q^{51} + 104 q^{52} - 112 q^{53} + 6 q^{54} - 516 q^{55} + 540 q^{56} - 378 q^{57} + 556 q^{58} - 372 q^{59} + 336 q^{60} - 1248 q^{61} + 612 q^{62} - 462 q^{63} + 680 q^{64} - 868 q^{65} - 360 q^{66} - 228 q^{67} + 80 q^{68} - 282 q^{69} + 192 q^{70} + 168 q^{71} - 300 q^{72} + 508 q^{73} + 104 q^{74} + 150 q^{75} + 276 q^{76} - 192 q^{77} + 96 q^{78} + 116 q^{79} - 64 q^{80} + 168 q^{81} + 220 q^{82} - 252 q^{83} + 384 q^{84} - 412 q^{85} - 480 q^{86} + 372 q^{87} - 912 q^{88} - 712 q^{89} + 24 q^{90} + 96 q^{91} - 1176 q^{92} + 174 q^{93} - 1356 q^{94} + 240 q^{95} - 228 q^{96} + 552 q^{97} - 536 q^{98} + 306 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(156))$$

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
156.3.d $$\chi_{156}(53, \cdot)$$ 156.3.d.a 8 1
156.3.e $$\chi_{156}(103, \cdot)$$ 156.3.e.a 2 1
156.3.e.b 2
156.3.e.c 24
156.3.f $$\chi_{156}(79, \cdot)$$ 156.3.f.a 24 1
156.3.g $$\chi_{156}(77, \cdot)$$ 156.3.g.a 2 1
156.3.g.b 4
156.3.g.c 4
156.3.j $$\chi_{156}(73, \cdot)$$ 156.3.j.a 8 2
156.3.l $$\chi_{156}(47, \cdot)$$ 156.3.l.a 4 2
156.3.l.b 4
156.3.l.c 96
156.3.n $$\chi_{156}(43, \cdot)$$ 156.3.n.a 2 2
156.3.n.b 2
156.3.n.c 4
156.3.n.d 4
156.3.n.e 22
156.3.n.f 22
156.3.o $$\chi_{156}(29, \cdot)$$ 156.3.o.a 2 2
156.3.o.b 16
156.3.s $$\chi_{156}(17, \cdot)$$ 156.3.s.a 2 2
156.3.s.b 16
156.3.t $$\chi_{156}(55, \cdot)$$ 156.3.t.a 28 2
156.3.t.b 28
156.3.v $$\chi_{156}(11, \cdot)$$ 156.3.v.a 208 4
156.3.x $$\chi_{156}(37, \cdot)$$ 156.3.x.a 8 4
156.3.x.b 12

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(156))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(156)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 2}$$