## Defining parameters

 Level: $$N$$ = $$135 = 3^{3} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$9$$ Newform subspaces: $$22$$ Sturm bound: $$5184$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(135))$$.

Total New Old
Modular forms 2064 1456 608
Cusp forms 1824 1360 464
Eisenstein series 240 96 144

## Trace form

 $$1360 q - 10 q^{2} - 12 q^{3} - 50 q^{4} - 37 q^{5} - 12 q^{6} + 54 q^{7} + 234 q^{8} + 84 q^{9} + O(q^{10})$$ $$1360 q - 10 q^{2} - 12 q^{3} - 50 q^{4} - 37 q^{5} - 12 q^{6} + 54 q^{7} + 234 q^{8} + 84 q^{9} - 31 q^{10} - 262 q^{11} - 318 q^{12} - 138 q^{13} - 18 q^{14} + 12 q^{15} + 546 q^{16} + 890 q^{17} + 1254 q^{18} + 114 q^{19} + 81 q^{20} + 228 q^{21} - 850 q^{22} - 1326 q^{23} - 2364 q^{24} - 59 q^{25} - 4372 q^{26} - 2274 q^{27} - 116 q^{28} - 182 q^{29} - 483 q^{30} + 1250 q^{31} + 4294 q^{32} + 2310 q^{33} + 2390 q^{34} + 3983 q^{35} + 6144 q^{36} + 1422 q^{37} + 2678 q^{38} + 672 q^{39} - 3007 q^{40} - 218 q^{41} + 2484 q^{42} - 3066 q^{43} - 5810 q^{44} - 2304 q^{45} - 4802 q^{46} - 8134 q^{47} - 8442 q^{48} - 2854 q^{49} - 6676 q^{50} - 5454 q^{51} + 1326 q^{52} - 1792 q^{53} - 7620 q^{54} + 2552 q^{55} + 702 q^{56} - 2844 q^{57} + 3894 q^{58} - 604 q^{59} + 1254 q^{60} - 742 q^{61} + 5856 q^{62} + 5196 q^{63} + 5682 q^{64} + 7881 q^{65} + 22182 q^{66} + 6738 q^{67} + 23044 q^{68} + 12252 q^{69} + 4845 q^{70} + 11794 q^{71} + 6192 q^{72} + 2598 q^{73} + 11134 q^{74} + 7584 q^{75} - 46 q^{76} + 2394 q^{77} - 5448 q^{78} - 4578 q^{79} - 7686 q^{80} - 8556 q^{81} - 8092 q^{82} - 17922 q^{83} - 28824 q^{84} - 11947 q^{85} - 39754 q^{86} - 15744 q^{87} - 27858 q^{88} - 17976 q^{89} - 7806 q^{90} - 12294 q^{91} + 762 q^{92} + 12492 q^{93} + 5726 q^{94} - 1623 q^{95} + 17988 q^{96} + 7854 q^{97} + 2188 q^{98} + 2580 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(135))$$

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
135.4.a $$\chi_{135}(1, \cdot)$$ 135.4.a.a 1 1
135.4.a.b 1
135.4.a.c 1
135.4.a.d 1
135.4.a.e 3
135.4.a.f 3
135.4.a.g 3
135.4.a.h 3
135.4.b $$\chi_{135}(109, \cdot)$$ 135.4.b.a 4 1
135.4.b.b 8
135.4.b.c 12
135.4.e $$\chi_{135}(46, \cdot)$$ 135.4.e.a 4 2
135.4.e.b 6
135.4.e.c 14
135.4.f $$\chi_{135}(53, \cdot)$$ 135.4.f.a 24 2
135.4.f.b 24
135.4.j $$\chi_{135}(19, \cdot)$$ 135.4.j.a 32 2
135.4.k $$\chi_{135}(16, \cdot)$$ 135.4.k.a 102 6
135.4.k.b 114
135.4.m $$\chi_{135}(8, \cdot)$$ 135.4.m.a 64 4
135.4.p $$\chi_{135}(4, \cdot)$$ 135.4.p.a 312 6
135.4.q $$\chi_{135}(2, \cdot)$$ 135.4.q.a 624 12

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(135))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(135)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$