## Defining parameters

 Level: $$N$$ = $$468 = 2^{2} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$10$$ Newform subspaces: $$12$$ Sturm bound: $$12096$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(468))$$.

Total New Old
Modular forms 546 138 408
Cusp forms 66 40 26
Eisenstein series 480 98 382

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 28 8 4 0

## Trace form

 $$40q + q^{2} + q^{3} - q^{4} + 6q^{5} + 2q^{6} - 2q^{8} - 3q^{9} + O(q^{10})$$ $$40q + q^{2} + q^{3} - q^{4} + 6q^{5} + 2q^{6} - 2q^{8} - 3q^{9} - 5q^{10} + 2q^{13} + 4q^{14} + 3q^{16} - 5q^{17} + 2q^{19} + q^{20} - 4q^{21} - 4q^{22} - 3q^{23} - 2q^{24} - 7q^{25} + q^{26} - 2q^{27} - q^{29} - 2q^{30} + 2q^{31} + q^{32} - 2q^{34} + 2q^{36} - 11q^{37} + 4q^{38} - q^{39} - 2q^{40} - 3q^{41} - 7q^{43} + 2q^{45} - 4q^{46} - 10q^{49} + 3q^{51} - 8q^{52} - 14q^{53} - 2q^{54} - 4q^{55} + 2q^{56} - 4q^{57} - 5q^{58} - 10q^{61} - 4q^{62} - 10q^{64} - 3q^{65} - 6q^{67} - 3q^{68} + q^{69} - 4q^{70} - 4q^{73} - q^{74} + 2q^{75} + 2q^{78} - 5q^{79} + 3q^{80} + q^{81} - 5q^{82} - 2q^{84} - 9q^{85} - 4q^{86} - 4q^{88} + 6q^{89} + 4q^{91} + 4q^{93} - 6q^{94} + 8q^{96} - 4q^{97} + q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(468))$$

We only show spaces with odd 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
468.1.d $$\chi_{468}(53, \cdot)$$ None 0 1
468.1.e $$\chi_{468}(415, \cdot)$$ 468.1.e.a 1 1
468.1.e.b 1
468.1.e.c 2
468.1.f $$\chi_{468}(235, \cdot)$$ None 0 1
468.1.g $$\chi_{468}(233, \cdot)$$ None 0 1
468.1.m $$\chi_{468}(73, \cdot)$$ None 0 2
468.1.o $$\chi_{468}(359, \cdot)$$ 468.1.o.a 4 2
468.1.q $$\chi_{468}(127, \cdot)$$ 468.1.q.a 4 2
468.1.r $$\chi_{468}(269, \cdot)$$ 468.1.r.a 4 2
468.1.u $$\chi_{468}(173, \cdot)$$ None 0 2
468.1.v $$\chi_{468}(211, \cdot)$$ 468.1.v.a 4 2
468.1.y $$\chi_{468}(139, \cdot)$$ 468.1.y.a 4 2
468.1.z $$\chi_{468}(77, \cdot)$$ 468.1.z.a 2 2
468.1.ba $$\chi_{468}(79, \cdot)$$ None 0 2
468.1.bb $$\chi_{468}(101, \cdot)$$ None 0 2
468.1.bf $$\chi_{468}(29, \cdot)$$ None 0 2
468.1.bg $$\chi_{468}(103, \cdot)$$ None 0 2
468.1.bh $$\chi_{468}(209, \cdot)$$ None 0 2
468.1.bi $$\chi_{468}(355, \cdot)$$ None 0 2
468.1.bn $$\chi_{468}(43, \cdot)$$ None 0 2
468.1.bo $$\chi_{468}(185, \cdot)$$ None 0 2
468.1.bq $$\chi_{468}(17, \cdot)$$ None 0 2
468.1.br $$\chi_{468}(55, \cdot)$$ 468.1.br.a 2 2
468.1.bt $$\chi_{468}(229, \cdot)$$ None 0 4
468.1.bu $$\chi_{468}(167, \cdot)$$ None 0 4
468.1.bx $$\chi_{468}(71, \cdot)$$ 468.1.bx.a 8 4
468.1.by $$\chi_{468}(11, \cdot)$$ None 0 4
468.1.ca $$\chi_{468}(97, \cdot)$$ None 0 4
468.1.cd $$\chi_{468}(37, \cdot)$$ 468.1.cd.a 4 4
468.1.ce $$\chi_{468}(85, \cdot)$$ None 0 4
468.1.ch $$\chi_{468}(47, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(468))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(468)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(117))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(156))$$$$^{\oplus 2}$$