## Defining parameters

 Level: $$N$$ = $$126 = 2 \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$10$$ Newform subspaces: $$28$$ Sturm bound: $$3456$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(126))$$.

Total New Old
Modular forms 1392 328 1064
Cusp forms 1200 328 872
Eisenstein series 192 0 192

## Trace form

 $$328q - 8q^{2} - 6q^{3} + 16q^{4} - 48q^{5} + 36q^{6} - 44q^{7} + 16q^{8} + 210q^{9} + O(q^{10})$$ $$328q - 8q^{2} - 6q^{3} + 16q^{4} - 48q^{5} + 36q^{6} - 44q^{7} + 16q^{8} + 210q^{9} + 60q^{10} - 12q^{11} - 48q^{12} - 250q^{13} - 464q^{14} - 420q^{15} + 64q^{16} + 330q^{17} + 24q^{18} + 488q^{19} + 264q^{20} + 756q^{21} + 732q^{22} + 1578q^{23} + 144q^{24} + 826q^{25} + 500q^{26} - 792q^{27} - 8q^{28} - 1164q^{29} - 264q^{30} - 1978q^{31} - 128q^{32} - 2274q^{33} - 1092q^{34} - 1920q^{35} - 1032q^{36} + 158q^{37} - 616q^{38} + 2424q^{39} + 240q^{40} + 3642q^{41} + 816q^{42} + 1202q^{43} + 888q^{44} + 2796q^{45} - 504q^{46} + 1902q^{47} + 288q^{48} - 3224q^{49} + 1408q^{50} + 3582q^{51} - 760q^{52} - 2046q^{53} - 1188q^{54} + 900q^{55} - 224q^{56} - 5046q^{57} + 1632q^{58} - 3930q^{59} - 2160q^{60} + 5720q^{61} + 824q^{62} - 8622q^{63} - 512q^{64} - 7092q^{65} - 5376q^{66} - 4648q^{67} - 3072q^{68} - 6912q^{69} - 804q^{70} - 1608q^{71} + 48q^{72} - 3034q^{73} + 1304q^{74} + 5214q^{75} - 256q^{76} + 8622q^{77} + 8280q^{78} + 4382q^{79} + 960q^{80} + 294q^{81} + 5016q^{82} + 5262q^{83} - 744q^{84} + 5424q^{85} - 4492q^{86} + 1692q^{87} - 2352q^{88} - 4950q^{89} - 4224q^{90} - 6970q^{91} + 528q^{92} + 2748q^{93} - 3000q^{94} + 1374q^{95} - 384q^{96} - 2470q^{97} + 3172q^{98} + 1032q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(126))$$

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
126.4.a $$\chi_{126}(1, \cdot)$$ 126.4.a.a 1 1
126.4.a.b 1
126.4.a.c 1
126.4.a.d 1
126.4.a.e 1
126.4.a.f 1
126.4.a.g 1
126.4.a.h 1
126.4.d $$\chi_{126}(125, \cdot)$$ 126.4.d.a 8 1
126.4.e $$\chi_{126}(25, \cdot)$$ 126.4.e.a 24 2
126.4.e.b 24
126.4.f $$\chi_{126}(43, \cdot)$$ 126.4.f.a 6 2
126.4.f.b 8
126.4.f.c 10
126.4.f.d 12
126.4.g $$\chi_{126}(37, \cdot)$$ 126.4.g.a 2 2
126.4.g.b 2
126.4.g.c 2
126.4.g.d 2
126.4.g.e 4
126.4.g.f 4
126.4.g.g 4
126.4.h $$\chi_{126}(67, \cdot)$$ 126.4.h.a 24 2
126.4.h.b 24
126.4.k $$\chi_{126}(17, \cdot)$$ 126.4.k.a 16 2
126.4.l $$\chi_{126}(5, \cdot)$$ 126.4.l.a 48 2
126.4.m $$\chi_{126}(41, \cdot)$$ 126.4.m.a 48 2
126.4.t $$\chi_{126}(47, \cdot)$$ 126.4.t.a 48 2

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(126))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(126)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 2}$$