## Defining parameters

 Level: $$N$$ = $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$8$$ Sturm bound: $$21504$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(624))$$.

Total New Old
Modular forms 748 123 625
Cusp forms 76 25 51
Eisenstein series 672 98 574

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 25 0 0 0

## Trace form

 $$25 q + q^{3} + 2 q^{7} - q^{9} + O(q^{10})$$ $$25 q + q^{3} + 2 q^{7} - q^{9} + 8 q^{12} - q^{13} + 2 q^{19} - 2 q^{21} + 8 q^{22} - q^{25} + q^{27} - 8 q^{30} + 2 q^{31} - 2 q^{37} - 5 q^{39} - 12 q^{43} - 17 q^{49} - 8 q^{52} - 8 q^{57} - 2 q^{61} - 4 q^{63} - 4 q^{67} - 2 q^{73} + 9 q^{75} + 2 q^{79} - 9 q^{81} - 8 q^{88} - 8 q^{90} + 2 q^{91} - 2 q^{93} - 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(624))$$

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
624.1.b $$\chi_{624}(233, \cdot)$$ None 0 1
624.1.e $$\chi_{624}(391, \cdot)$$ None 0 1
624.1.f $$\chi_{624}(209, \cdot)$$ None 0 1
624.1.i $$\chi_{624}(415, \cdot)$$ None 0 1
624.1.k $$\chi_{624}(79, \cdot)$$ None 0 1
624.1.l $$\chi_{624}(545, \cdot)$$ 624.1.l.a 1 1
624.1.o $$\chi_{624}(103, \cdot)$$ None 0 1
624.1.p $$\chi_{624}(521, \cdot)$$ None 0 1
624.1.s $$\chi_{624}(395, \cdot)$$ None 0 2
624.1.t $$\chi_{624}(109, \cdot)$$ None 0 2
624.1.w $$\chi_{624}(53, \cdot)$$ None 0 2
624.1.y $$\chi_{624}(259, \cdot)$$ None 0 2
624.1.z $$\chi_{624}(73, \cdot)$$ None 0 2
624.1.ba $$\chi_{624}(385, \cdot)$$ None 0 2
624.1.bd $$\chi_{624}(47, \cdot)$$ 624.1.bd.a 2 2
624.1.bd.b 2
624.1.be $$\chi_{624}(359, \cdot)$$ None 0 2
624.1.bi $$\chi_{624}(77, \cdot)$$ 624.1.bi.a 8 2
624.1.bk $$\chi_{624}(235, \cdot)$$ None 0 2
624.1.bl $$\chi_{624}(421, \cdot)$$ None 0 2
624.1.bo $$\chi_{624}(83, \cdot)$$ None 0 2
624.1.bp $$\chi_{624}(127, \cdot)$$ None 0 2
624.1.bs $$\chi_{624}(113, \cdot)$$ 624.1.bs.a 2 2
624.1.bt $$\chi_{624}(55, \cdot)$$ None 0 2
624.1.bw $$\chi_{624}(329, \cdot)$$ None 0 2
624.1.bx $$\chi_{624}(185, \cdot)$$ None 0 2
624.1.by $$\chi_{624}(199, \cdot)$$ None 0 2
624.1.cb $$\chi_{624}(17, \cdot)$$ 624.1.cb.a 2 2
624.1.cc $$\chi_{624}(367, \cdot)$$ None 0 2
624.1.cf $$\chi_{624}(37, \cdot)$$ None 0 4
624.1.cg $$\chi_{624}(323, \cdot)$$ None 0 4
624.1.ci $$\chi_{624}(139, \cdot)$$ None 0 4
624.1.ck $$\chi_{624}(101, \cdot)$$ None 0 4
624.1.co $$\chi_{624}(71, \cdot)$$ None 0 4
624.1.cp $$\chi_{624}(383, \cdot)$$ 624.1.cp.a 4 4
624.1.cp.b 4
624.1.cs $$\chi_{624}(97, \cdot)$$ None 0 4
624.1.ct $$\chi_{624}(409, \cdot)$$ None 0 4
624.1.cu $$\chi_{624}(43, \cdot)$$ None 0 4
624.1.cw $$\chi_{624}(29, \cdot)$$ None 0 4
624.1.cy $$\chi_{624}(11, \cdot)$$ None 0 4
624.1.db $$\chi_{624}(349, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(624))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(624)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 5}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(156))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(208))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(312))$$$$^{\oplus 2}$$