## Defining parameters

 Level: $$N$$ = $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$16$$ Newform subspaces: $$32$$ Sturm bound: $$6480$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(171))$$.

Total New Old
Modular forms 2304 1779 525
Cusp forms 2016 1625 391
Eisenstein series 288 154 134

## Trace form

 $$1625 q - 21 q^{2} - 30 q^{3} - 25 q^{4} - 39 q^{5} - 54 q^{6} - 23 q^{7} - 27 q^{8} - 18 q^{9} + O(q^{10})$$ $$1625 q - 21 q^{2} - 30 q^{3} - 25 q^{4} - 39 q^{5} - 54 q^{6} - 23 q^{7} - 27 q^{8} - 18 q^{9} - 57 q^{10} - 21 q^{11} - 48 q^{12} + 25 q^{13} + 33 q^{14} - 36 q^{15} + 23 q^{16} - 18 q^{17} - 36 q^{18} - 124 q^{19} - 114 q^{20} - 24 q^{21} - 141 q^{22} + 24 q^{23} + 54 q^{24} - 127 q^{25} - 171 q^{26} - 144 q^{27} + 34 q^{28} - 39 q^{29} - 72 q^{30} + 145 q^{31} + 144 q^{32} - 54 q^{33} + 162 q^{34} + 45 q^{35} - 90 q^{36} + 64 q^{37} - 57 q^{38} - 24 q^{39} - 312 q^{40} - 57 q^{41} - 36 q^{42} - 482 q^{43} - 1314 q^{44} - 522 q^{45} - 1758 q^{46} - 768 q^{47} - 1110 q^{48} - 774 q^{49} - 1752 q^{50} - 648 q^{51} - 827 q^{52} - 351 q^{53} - 90 q^{54} - 165 q^{55} + 6 q^{56} + 87 q^{57} + 318 q^{58} + 204 q^{59} + 792 q^{60} + 1045 q^{61} + 1530 q^{62} + 396 q^{63} + 2579 q^{64} + 1680 q^{65} + 1440 q^{66} + 1180 q^{67} + 2736 q^{68} + 666 q^{69} + 2001 q^{70} + 1332 q^{71} + 558 q^{72} - 26 q^{73} - 267 q^{74} + 42 q^{75} - 106 q^{76} - 93 q^{77} - 108 q^{78} - 17 q^{79} + 801 q^{80} + 126 q^{81} - 678 q^{82} + 150 q^{83} - 24 q^{84} - 315 q^{85} - 66 q^{86} + 432 q^{87} - 453 q^{88} - 648 q^{89} - 594 q^{90} - 709 q^{91} - 744 q^{92} - 420 q^{93} - 1554 q^{94} - 2064 q^{95} - 2520 q^{96} - 2066 q^{97} - 4770 q^{98} - 1800 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(171))$$

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
171.3.b $$\chi_{171}(134, \cdot)$$ 171.3.b.a 12 1
171.3.c $$\chi_{171}(37, \cdot)$$ 171.3.c.a 1 1
171.3.c.b 2
171.3.c.c 2
171.3.c.d 2
171.3.c.e 4
171.3.c.f 4
171.3.i $$\chi_{171}(88, \cdot)$$ 171.3.i.a 76 2
171.3.j $$\chi_{171}(68, \cdot)$$ 171.3.j.a 76 2
171.3.n $$\chi_{171}(11, \cdot)$$ 171.3.n.a 76 2
171.3.o $$\chi_{171}(94, \cdot)$$ 171.3.o.a 76 2
171.3.p $$\chi_{171}(46, \cdot)$$ 171.3.p.a 2 2
171.3.p.b 2
171.3.p.c 6
171.3.p.d 6
171.3.p.e 6
171.3.p.f 8
171.3.q $$\chi_{171}(20, \cdot)$$ 171.3.q.a 72 2
171.3.r $$\chi_{171}(26, \cdot)$$ 171.3.r.a 4 2
171.3.r.b 4
171.3.r.c 16
171.3.s $$\chi_{171}(31, \cdot)$$ 171.3.s.a 76 2
171.3.z $$\chi_{171}(5, \cdot)$$ 171.3.z.a 228 6
171.3.ba $$\chi_{171}(10, \cdot)$$ 171.3.ba.a 6 6
171.3.ba.b 12
171.3.ba.c 18
171.3.ba.d 24
171.3.ba.e 36
171.3.bb $$\chi_{171}(17, \cdot)$$ 171.3.bb.a 84 6
171.3.bc $$\chi_{171}(22, \cdot)$$ 171.3.bc.a 228 6
171.3.be $$\chi_{171}(13, \cdot)$$ 171.3.be.a 228 6
171.3.bf $$\chi_{171}(23, \cdot)$$ 171.3.bf.a 228 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(171))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(171)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 2}$$