## Defining parameters

 Level: $$N$$ = $$234 = 2 \cdot 3^{2} \cdot 13$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$15$$ Newform subspaces: $$32$$ Sturm bound: $$9072$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(234))$$.

Total New Old
Modular forms 3216 786 2430
Cusp forms 2832 786 2046
Eisenstein series 384 0 384

## Trace form

 $$786 q + 36 q^{5} + 24 q^{6} - 8 q^{7} - 12 q^{8} - 24 q^{9} + O(q^{10})$$ $$786 q + 36 q^{5} + 24 q^{6} - 8 q^{7} - 12 q^{8} - 24 q^{9} - 54 q^{10} - 48 q^{11} - 24 q^{12} + 18 q^{13} - 48 q^{14} - 36 q^{15} + 16 q^{16} + 36 q^{17} + 48 q^{18} + 184 q^{19} + 96 q^{20} + 84 q^{21} + 120 q^{22} - 60 q^{23} - 36 q^{25} + 72 q^{28} + 672 q^{29} + 240 q^{30} + 408 q^{31} + 252 q^{33} + 168 q^{34} + 660 q^{35} + 72 q^{36} + 144 q^{38} + 6 q^{39} + 48 q^{40} - 210 q^{41} - 192 q^{42} - 296 q^{43} - 432 q^{44} - 732 q^{45} - 624 q^{46} - 1128 q^{47} - 48 q^{48} - 776 q^{49} - 906 q^{50} - 480 q^{51} - 192 q^{52} - 240 q^{53} - 72 q^{54} - 300 q^{55} - 240 q^{56} - 288 q^{57} - 186 q^{58} - 336 q^{59} + 72 q^{60} - 246 q^{61} - 72 q^{62} + 228 q^{63} + 96 q^{64} + 1296 q^{65} + 144 q^{66} + 1108 q^{67} - 132 q^{68} + 540 q^{69} + 528 q^{70} + 1032 q^{71} + 192 q^{72} + 1312 q^{73} - 66 q^{74} + 264 q^{75} - 128 q^{76} + 180 q^{77} - 144 q^{78} - 36 q^{79} + 48 q^{80} + 120 q^{81} - 354 q^{82} - 576 q^{83} + 288 q^{84} - 1566 q^{85} + 120 q^{86} - 852 q^{87} - 240 q^{88} - 1512 q^{89} - 1304 q^{91} - 24 q^{92} - 1596 q^{93} - 408 q^{94} - 1416 q^{95} - 96 q^{96} + 156 q^{97} - 1104 q^{98} - 2700 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(234))$$

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
234.3.c $$\chi_{234}(53, \cdot)$$ 234.3.c.a 4 1
234.3.c.b 4
234.3.d $$\chi_{234}(233, \cdot)$$ 234.3.d.a 4 1
234.3.d.b 4
234.3.d.c 4
234.3.i $$\chi_{234}(73, \cdot)$$ 234.3.i.a 2 2
234.3.i.b 4
234.3.i.c 6
234.3.i.d 6
234.3.i.e 8
234.3.k $$\chi_{234}(35, \cdot)$$ 234.3.k.a 4 2
234.3.k.b 12
234.3.m $$\chi_{234}(23, \cdot)$$ 234.3.m.a 56 2
234.3.n $$\chi_{234}(77, \cdot)$$ 234.3.n.a 56 2
234.3.o $$\chi_{234}(95, \cdot)$$ 234.3.o.a 56 2
234.3.q $$\chi_{234}(29, \cdot)$$ 234.3.q.a 56 2
234.3.r $$\chi_{234}(131, \cdot)$$ 234.3.r.a 48 2
234.3.u $$\chi_{234}(185, \cdot)$$ 234.3.u.a 56 2
234.3.v $$\chi_{234}(17, \cdot)$$ 234.3.v.a 16 2
234.3.w $$\chi_{234}(31, \cdot)$$ 234.3.w.a 56 4
234.3.w.b 56
234.3.ba $$\chi_{234}(7, \cdot)$$ 234.3.ba.a 56 4
234.3.ba.b 56
234.3.bb $$\chi_{234}(19, \cdot)$$ 234.3.bb.a 4 4
234.3.bb.b 4
234.3.bb.c 4
234.3.bb.d 8
234.3.bb.e 8
234.3.bb.f 8
234.3.bb.g 8
234.3.bc $$\chi_{234}(85, \cdot)$$ 234.3.bc.a 56 4
234.3.bc.b 56

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(234))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(234)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(117))$$$$^{\oplus 2}$$