## Defining parameters

 Level: $$N$$ = $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newform subspaces: $$32$$ Sturm bound: $$4320$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(171))$$.

Total New Old
Modular forms 1224 926 298
Cusp forms 937 774 163
Eisenstein series 287 152 135

## Trace form

 $$774q - 27q^{2} - 36q^{3} - 27q^{4} - 27q^{5} - 36q^{6} - 27q^{7} - 27q^{8} - 36q^{9} + O(q^{10})$$ $$774q - 27q^{2} - 36q^{3} - 27q^{4} - 27q^{5} - 36q^{6} - 27q^{7} - 27q^{8} - 36q^{9} - 81q^{10} - 27q^{11} - 36q^{12} - 39q^{13} - 45q^{14} - 36q^{15} - 63q^{16} - 36q^{17} - 36q^{18} - 102q^{19} - 90q^{20} - 36q^{21} - 54q^{22} - 36q^{23} - 36q^{24} - 45q^{25} - 45q^{26} - 36q^{27} - 120q^{28} - 45q^{29} - 36q^{30} - 45q^{31} - 72q^{32} - 36q^{33} - 72q^{34} - 63q^{35} - 108q^{37} - 90q^{38} - 72q^{39} - 108q^{40} - 45q^{41} - 36q^{42} - 54q^{43} + 9q^{44} + 18q^{45} - 18q^{46} + 36q^{47} + 90q^{48} + 18q^{49} + 162q^{50} + 54q^{51} + 171q^{52} + 81q^{53} + 72q^{54} + 27q^{55} + 324q^{56} + 54q^{57} + 54q^{58} + 108q^{59} + 180q^{60} + 51q^{61} + 90q^{62} + 36q^{63} + 45q^{64} + 54q^{65} + 108q^{66} - 36q^{67} + 99q^{68} + 18q^{69} - 27q^{70} + 18q^{71} + 18q^{72} - 153q^{73} - 45q^{74} - 36q^{75} - 117q^{76} - 153q^{77} - 36q^{78} - 129q^{79} - 45q^{80} - 36q^{81} - 252q^{82} - 90q^{83} - 36q^{84} - 135q^{85} - 126q^{86} - 36q^{87} - 189q^{88} - 108q^{89} + 54q^{90} - 165q^{91} - 144q^{92} - 36q^{93} - 72q^{94} + 54q^{95} + 72q^{96} + 54q^{97} + 198q^{98} + 90q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(171))$$

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
171.2.a $$\chi_{171}(1, \cdot)$$ 171.2.a.a 1 1
171.2.a.b 1
171.2.a.c 1
171.2.a.d 1
171.2.a.e 4
171.2.d $$\chi_{171}(170, \cdot)$$ 171.2.d.a 4 1
171.2.d.b 4
171.2.e $$\chi_{171}(58, \cdot)$$ 171.2.e.a 18 2
171.2.e.b 18
171.2.f $$\chi_{171}(64, \cdot)$$ 171.2.f.a 2 2
171.2.f.b 6
171.2.f.c 8
171.2.g $$\chi_{171}(106, \cdot)$$ 171.2.g.a 2 2
171.2.g.b 2
171.2.g.c 32
171.2.h $$\chi_{171}(7, \cdot)$$ 171.2.h.a 2 2
171.2.h.b 2
171.2.h.c 32
171.2.k $$\chi_{171}(50, \cdot)$$ 171.2.k.a 36 2
171.2.l $$\chi_{171}(56, \cdot)$$ 171.2.l.a 36 2
171.2.m $$\chi_{171}(8, \cdot)$$ 171.2.m.a 16 2
171.2.t $$\chi_{171}(122, \cdot)$$ 171.2.t.a 36 2
171.2.u $$\chi_{171}(28, \cdot)$$ 171.2.u.a 6 6
171.2.u.b 6
171.2.u.c 6
171.2.u.d 12
171.2.u.e 12
171.2.v $$\chi_{171}(25, \cdot)$$ 171.2.v.a 108 6
171.2.w $$\chi_{171}(4, \cdot)$$ 171.2.w.a 108 6
171.2.x $$\chi_{171}(14, \cdot)$$ 171.2.x.a 108 6
171.2.y $$\chi_{171}(53, \cdot)$$ 171.2.y.a 36 6
171.2.bd $$\chi_{171}(2, \cdot)$$ 171.2.bd.a 108 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(171))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(171)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 2}$$