## Defining parameters

 Level: $$N$$ = $$1110 = 2 \cdot 3 \cdot 5 \cdot 37$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$30$$ Sturm bound: $$262656$$ Trace bound: $$8$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1110))$$.

Total New Old
Modular forms 99648 22540 77108
Cusp forms 97344 22540 74804
Eisenstein series 2304 0 2304

## Trace form

 $$22540q - 28q^{3} - 8q^{5} + 8q^{6} - 80q^{7} + O(q^{10})$$ $$22540q - 28q^{3} - 8q^{5} + 8q^{6} - 80q^{7} + 80q^{10} - 64q^{11} + 16q^{12} + 304q^{13} + 128q^{14} + 476q^{15} + 256q^{16} + 144q^{17} + 32q^{18} - 32q^{19} + 32q^{20} - 1120q^{21} - 672q^{22} - 144q^{23} - 96q^{24} - 640q^{25} + 2088q^{26} + 2564q^{27} + 1408q^{28} + 3312q^{29} + 264q^{30} - 2496q^{31} - 4384q^{33} - 8192q^{34} - 7792q^{35} - 2320q^{36} - 12664q^{37} - 5760q^{38} - 5496q^{39} - 1968q^{40} - 6160q^{41} + 1312q^{42} + 7024q^{43} + 1984q^{44} + 2728q^{45} + 18368q^{46} + 10944q^{47} + 2368q^{48} + 20976q^{49} + 2980q^{50} - 344q^{51} - 704q^{52} - 288q^{53} - 216q^{54} - 3584q^{55} + 640q^{56} - 2320q^{57} - 5376q^{58} + 16448q^{59} - 2320q^{60} + 13820q^{61} - 2016q^{62} + 4952q^{63} - 2178q^{65} - 352q^{66} - 3776q^{67} + 576q^{68} - 14496q^{69} + 5760q^{70} - 15840q^{71} - 128q^{72} - 21920q^{73} - 80q^{74} - 10328q^{75} + 1408q^{76} - 14112q^{77} + 7728q^{78} - 8240q^{79} - 128q^{80} + 3088q^{81} - 2688q^{82} + 4752q^{83} - 576q^{84} + 10774q^{85} - 2912q^{86} + 15008q^{87} - 2688q^{88} + 27244q^{89} - 10768q^{90} + 28016q^{91} - 576q^{92} - 43744q^{93} - 4864q^{94} + 2352q^{95} + 640q^{96} + 17872q^{97} + 4032q^{98} - 3456q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1110))$$

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
1110.4.a $$\chi_{1110}(1, \cdot)$$ 1110.4.a.a 1 1
1110.4.a.b 1
1110.4.a.c 2
1110.4.a.d 2
1110.4.a.e 3
1110.4.a.f 3
1110.4.a.g 3
1110.4.a.h 3
1110.4.a.i 4
1110.4.a.j 4
1110.4.a.k 4
1110.4.a.l 4
1110.4.a.m 5
1110.4.a.n 5
1110.4.a.o 5
1110.4.a.p 5
1110.4.a.q 5
1110.4.a.r 6
1110.4.a.s 7
1110.4.d $$\chi_{1110}(889, \cdot)$$ n/a 108 1
1110.4.e $$\chi_{1110}(739, \cdot)$$ n/a 116 1
1110.4.h $$\chi_{1110}(961, \cdot)$$ 1110.4.h.a 4 1
1110.4.h.b 12
1110.4.h.c 20
1110.4.h.d 20
1110.4.h.e 20
1110.4.i $$\chi_{1110}(121, \cdot)$$ n/a 152 2
1110.4.k $$\chi_{1110}(179, \cdot)$$ n/a 456 2
1110.4.l $$\chi_{1110}(43, \cdot)$$ n/a 228 2
1110.4.m $$\chi_{1110}(593, \cdot)$$ n/a 432 2
1110.4.n $$\chi_{1110}(443, \cdot)$$ n/a 456 2
1110.4.o $$\chi_{1110}(253, \cdot)$$ n/a 228 2
1110.4.u $$\chi_{1110}(191, \cdot)$$ n/a 304 2
1110.4.x $$\chi_{1110}(751, \cdot)$$ n/a 152 2
1110.4.ba $$\chi_{1110}(529, \cdot)$$ n/a 232 2
1110.4.bb $$\chi_{1110}(1009, \cdot)$$ n/a 224 2
1110.4.bc $$\chi_{1110}(181, \cdot)$$ n/a 456 6
1110.4.be $$\chi_{1110}(251, \cdot)$$ n/a 608 4
1110.4.bf $$\chi_{1110}(97, \cdot)$$ n/a 456 4
1110.4.bg $$\chi_{1110}(233, \cdot)$$ n/a 912 4
1110.4.bh $$\chi_{1110}(47, \cdot)$$ n/a 912 4
1110.4.bi $$\chi_{1110}(193, \cdot)$$ n/a 456 4
1110.4.bo $$\chi_{1110}(29, \cdot)$$ n/a 912 4
1110.4.bp $$\chi_{1110}(139, \cdot)$$ n/a 696 6
1110.4.bq $$\chi_{1110}(49, \cdot)$$ n/a 672 6
1110.4.br $$\chi_{1110}(151, \cdot)$$ n/a 456 6
1110.4.by $$\chi_{1110}(59, \cdot)$$ n/a 2736 12
1110.4.bz $$\chi_{1110}(131, \cdot)$$ n/a 1824 12
1110.4.cc $$\chi_{1110}(163, \cdot)$$ n/a 1368 12
1110.4.cd $$\chi_{1110}(53, \cdot)$$ n/a 2736 12
1110.4.cg $$\chi_{1110}(77, \cdot)$$ n/a 2736 12
1110.4.ch $$\chi_{1110}(13, \cdot)$$ n/a 1368 12

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1110))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1110)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(30))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(37))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(74))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(111))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(185))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(222))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(370))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(555))$$$$^{\oplus 2}$$