## Defining parameters

 Level: $$N$$ = $$456 = 2^{3} \cdot 3 \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$47$$ Sturm bound: $$23040$$ Trace bound: $$6$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(456))$$.

Total New Old
Modular forms 6192 2484 3708
Cusp forms 5329 2348 2981
Eisenstein series 863 136 727

## Trace form

 $$2348q + 4q^{2} - 12q^{3} - 28q^{4} + 4q^{5} - 22q^{6} - 28q^{7} - 8q^{8} - 30q^{9} + O(q^{10})$$ $$2348q + 4q^{2} - 12q^{3} - 28q^{4} + 4q^{5} - 22q^{6} - 28q^{7} - 8q^{8} - 30q^{9} - 44q^{10} - 8q^{11} - 34q^{12} + 4q^{13} - 8q^{14} - 30q^{15} - 36q^{16} + 4q^{17} - 6q^{18} - 36q^{19} + 16q^{20} - 20q^{22} + 6q^{24} - 54q^{25} + 16q^{26} - 27q^{27} - 36q^{28} + 24q^{29} - 10q^{30} - 24q^{31} - 16q^{32} + 10q^{33} - 76q^{34} + 72q^{35} - 26q^{36} + 24q^{37} - 8q^{38} + 30q^{39} - 52q^{40} + 40q^{41} - 26q^{42} + 68q^{43} + 58q^{45} - 20q^{46} + 84q^{47} - 2q^{48} - 38q^{49} + 4q^{50} + 27q^{51} - 68q^{52} + 4q^{53} - 30q^{54} - 20q^{55} + 16q^{56} - 44q^{57} - 48q^{58} - 8q^{59} - 54q^{60} - 32q^{61} - 172q^{62} - 44q^{63} - 292q^{64} - 148q^{65} - 178q^{66} - 300q^{67} - 252q^{68} - 16q^{69} - 452q^{70} - 172q^{71} - 150q^{72} - 238q^{73} - 320q^{74} - 184q^{75} - 372q^{76} - 108q^{77} - 214q^{78} - 312q^{79} - 288q^{80} - 14q^{81} - 504q^{82} - 100q^{83} - 218q^{84} + 8q^{85} - 236q^{86} - 90q^{87} - 356q^{88} - 56q^{89} - 46q^{90} - 180q^{91} - 180q^{92} - 2q^{93} - 156q^{94} + 8q^{95} - 84q^{96} - 28q^{97} - 12q^{98} - 57q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(456))$$

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
456.2.a $$\chi_{456}(1, \cdot)$$ 456.2.a.a 1 1
456.2.a.b 1
456.2.a.c 1
456.2.a.d 1
456.2.a.e 2
456.2.a.f 2
456.2.d $$\chi_{456}(191, \cdot)$$ None 0 1
456.2.e $$\chi_{456}(379, \cdot)$$ 456.2.e.a 40 1
456.2.f $$\chi_{456}(113, \cdot)$$ 456.2.f.a 10 1
456.2.f.b 10
456.2.g $$\chi_{456}(229, \cdot)$$ 456.2.g.a 18 1
456.2.g.b 18
456.2.j $$\chi_{456}(419, \cdot)$$ 456.2.j.a 4 1
456.2.j.b 8
456.2.j.c 12
456.2.j.d 24
456.2.j.e 24
456.2.k $$\chi_{456}(151, \cdot)$$ None 0 1
456.2.p $$\chi_{456}(341, \cdot)$$ 456.2.p.a 12 1
456.2.p.b 64
456.2.q $$\chi_{456}(49, \cdot)$$ 456.2.q.a 2 2
456.2.q.b 2
456.2.q.c 2
456.2.q.d 4
456.2.q.e 4
456.2.q.f 6
456.2.t $$\chi_{456}(31, \cdot)$$ None 0 2
456.2.u $$\chi_{456}(11, \cdot)$$ 456.2.u.a 4 2
456.2.u.b 4
456.2.u.c 8
456.2.u.d 136
456.2.v $$\chi_{456}(221, \cdot)$$ 456.2.v.a 152 2
456.2.y $$\chi_{456}(259, \cdot)$$ 456.2.y.a 80 2
456.2.z $$\chi_{456}(239, \cdot)$$ None 0 2
456.2.be $$\chi_{456}(277, \cdot)$$ 456.2.be.a 80 2
456.2.bf $$\chi_{456}(65, \cdot)$$ 456.2.bf.a 4 2
456.2.bf.b 4
456.2.bf.c 16
456.2.bf.d 16
456.2.bg $$\chi_{456}(25, \cdot)$$ 456.2.bg.a 12 6
456.2.bg.b 12
456.2.bg.c 18
456.2.bg.d 18
456.2.bj $$\chi_{456}(29, \cdot)$$ 456.2.bj.a 456 6
456.2.bk $$\chi_{456}(61, \cdot)$$ 456.2.bk.a 240 6
456.2.bm $$\chi_{456}(41, \cdot)$$ 456.2.bm.a 60 6
456.2.bm.b 60
456.2.bp $$\chi_{456}(67, \cdot)$$ 456.2.bp.a 240 6
456.2.br $$\chi_{456}(23, \cdot)$$ None 0 6
456.2.bs $$\chi_{456}(79, \cdot)$$ None 0 6
456.2.bu $$\chi_{456}(35, \cdot)$$ 456.2.bu.a 12 6
456.2.bu.b 12
456.2.bu.c 432

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(456))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(456)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(57))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(114))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(228))$$$$^{\oplus 2}$$