## Defining parameters

 Level: $$N$$ = $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$9$$ Newform subspaces: $$18$$ Sturm bound: $$3888$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(135))$$.

Total New Old
Modular forms 1416 986 430
Cusp forms 1176 890 286
Eisenstein series 240 96 144

## Trace form

 $$890 q - 10 q^{2} - 12 q^{3} - 2 q^{4} + 5 q^{5} - 12 q^{6} - 6 q^{7} + 18 q^{8} - 24 q^{9} + O(q^{10})$$ $$890 q - 10 q^{2} - 12 q^{3} - 2 q^{4} + 5 q^{5} - 12 q^{6} - 6 q^{7} + 18 q^{8} - 24 q^{9} + q^{10} - 2 q^{11} + 6 q^{12} + 26 q^{13} + 30 q^{14} - 15 q^{15} - 62 q^{16} - 34 q^{17} - 150 q^{18} - 64 q^{19} - 267 q^{20} - 312 q^{21} - 82 q^{22} - 292 q^{23} - 312 q^{24} - 43 q^{25} + 64 q^{26} + 156 q^{27} + 156 q^{28} + 480 q^{29} + 219 q^{30} - 6 q^{31} + 302 q^{32} + 258 q^{33} - 346 q^{34} - 448 q^{35} + 96 q^{36} - 356 q^{37} - 870 q^{38} - 354 q^{39} - 567 q^{40} - 866 q^{41} - 1188 q^{42} - 178 q^{43} - 1314 q^{44} - 441 q^{45} - 186 q^{46} - 490 q^{47} - 450 q^{48} + 454 q^{49} + 1190 q^{50} + 216 q^{51} + 838 q^{52} + 1124 q^{53} + 804 q^{54} + 442 q^{55} + 2430 q^{56} + 828 q^{57} + 694 q^{58} + 1470 q^{59} + 1254 q^{60} + 470 q^{61} + 1876 q^{62} + 1146 q^{63} + 74 q^{64} + 311 q^{65} + 798 q^{66} - 684 q^{67} - 752 q^{68} - 222 q^{69} - 1131 q^{70} - 926 q^{71} - 2232 q^{72} - 1054 q^{73} - 2970 q^{74} - 1137 q^{75} - 2366 q^{76} - 2618 q^{77} - 2100 q^{78} - 1290 q^{79} - 3088 q^{80} - 888 q^{81} - 1004 q^{82} - 1570 q^{83} + 12 q^{84} - 317 q^{85} - 1394 q^{86} + 294 q^{87} + 1134 q^{88} - 180 q^{89} - 84 q^{90} + 150 q^{91} + 1658 q^{92} - 630 q^{93} + 2654 q^{94} + 957 q^{95} + 924 q^{96} + 1720 q^{97} + 2140 q^{98} + 150 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(135))$$

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
135.3.c $$\chi_{135}(26, \cdot)$$ 135.3.c.a 2 1
135.3.c.b 4
135.3.c.c 4
135.3.d $$\chi_{135}(134, \cdot)$$ 135.3.d.a 2 1
135.3.d.b 2
135.3.d.c 2
135.3.d.d 2
135.3.d.e 2
135.3.d.f 2
135.3.d.g 4
135.3.g $$\chi_{135}(28, \cdot)$$ 135.3.g.a 16 2
135.3.g.b 16
135.3.h $$\chi_{135}(44, \cdot)$$ 135.3.h.a 20 2
135.3.i $$\chi_{135}(71, \cdot)$$ 135.3.i.a 16 2
135.3.l $$\chi_{135}(37, \cdot)$$ 135.3.l.a 40 4
135.3.n $$\chi_{135}(14, \cdot)$$ 135.3.n.a 204 6
135.3.o $$\chi_{135}(11, \cdot)$$ 135.3.o.a 144 6
135.3.r $$\chi_{135}(7, \cdot)$$ 135.3.r.a 408 12

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(135))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(135)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$