## Defining parameters

 Level: $$N$$ = $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$17$$ Sturm bound: $$1728$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(90))$$.

Total New Old
Modular forms 712 157 555
Cusp forms 584 157 427
Eisenstein series 128 0 128

## Trace form

 $$157 q - 6 q^{2} - 6 q^{3} + 12 q^{4} - 17 q^{5} + 36 q^{6} + 4 q^{7} + 24 q^{8} + 82 q^{9} + O(q^{10})$$ $$157 q - 6 q^{2} - 6 q^{3} + 12 q^{4} - 17 q^{5} + 36 q^{6} + 4 q^{7} + 24 q^{8} + 82 q^{9} - 34 q^{10} - 34 q^{11} + 16 q^{12} + 298 q^{13} + 200 q^{14} + 246 q^{15} + 112 q^{16} + 354 q^{17} - 280 q^{18} - 632 q^{19} - 244 q^{20} - 720 q^{21} - 564 q^{22} - 348 q^{23} + 48 q^{24} + 221 q^{25} + 564 q^{26} + 552 q^{27} + 304 q^{28} + 1878 q^{29} + 288 q^{30} + 948 q^{31} - 96 q^{32} + 286 q^{33} - 824 q^{34} + 128 q^{35} - 584 q^{36} - 2366 q^{37} - 972 q^{38} - 932 q^{39} + 120 q^{40} - 2128 q^{41} - 800 q^{42} + 1222 q^{43} + 640 q^{44} - 2818 q^{45} + 1712 q^{46} - 2148 q^{47} - 224 q^{48} + 303 q^{49} - 734 q^{50} - 30 q^{51} - 440 q^{52} + 954 q^{53} + 300 q^{54} - 248 q^{55} + 544 q^{56} + 4250 q^{57} - 1668 q^{58} + 1298 q^{59} - 152 q^{60} - 5566 q^{61} + 48 q^{62} + 2396 q^{63} - 960 q^{64} - 1068 q^{65} - 480 q^{66} - 4718 q^{67} + 480 q^{68} - 772 q^{69} + 3528 q^{70} + 7968 q^{71} + 48 q^{72} + 1618 q^{73} + 2180 q^{74} + 4740 q^{75} + 4408 q^{76} + 9000 q^{77} + 2216 q^{78} + 7900 q^{79} + 592 q^{80} + 842 q^{81} + 6276 q^{82} - 2292 q^{83} - 1872 q^{84} + 1258 q^{85} - 7484 q^{86} - 8360 q^{87} - 2256 q^{88} - 16454 q^{89} - 6944 q^{90} - 17824 q^{91} - 1392 q^{92} - 5836 q^{93} - 3304 q^{94} + 468 q^{95} - 512 q^{96} + 3196 q^{97} + 438 q^{98} + 2512 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(90))$$

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
90.4.a $$\chi_{90}(1, \cdot)$$ 90.4.a.a 1 1
90.4.a.b 1
90.4.a.c 1
90.4.a.d 1
90.4.a.e 1
90.4.c $$\chi_{90}(19, \cdot)$$ 90.4.c.a 2 1
90.4.c.b 2
90.4.c.c 4
90.4.e $$\chi_{90}(31, \cdot)$$ 90.4.e.a 2 2
90.4.e.b 4
90.4.e.c 4
90.4.e.d 6
90.4.e.e 8
90.4.f $$\chi_{90}(17, \cdot)$$ 90.4.f.a 4 2
90.4.f.b 8
90.4.i $$\chi_{90}(49, \cdot)$$ 90.4.i.a 36 2
90.4.l $$\chi_{90}(23, \cdot)$$ 90.4.l.a 72 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(90))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(90)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$