## Defining parameters

 Level: $$N$$ = $$504 = 2^{3} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$7$$ Newform subspaces: $$10$$ Sturm bound: $$13824$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(504))$$.

Total New Old
Modular forms 646 121 525
Cusp forms 70 31 39
Eisenstein series 576 90 486

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 19 8 4 0

## Trace form

 $$31q + q^{2} - 2q^{3} - 5q^{4} + q^{7} + q^{8} + 2q^{9} + O(q^{10})$$ $$31q + q^{2} - 2q^{3} - 5q^{4} + q^{7} + q^{8} + 2q^{9} - 2q^{10} - 2q^{11} + 2q^{12} - 4q^{13} - 3q^{14} + 8q^{15} - 5q^{16} + 4q^{17} - 4q^{18} - 2q^{19} + 4q^{22} + 2q^{23} - 5q^{25} + 4q^{26} - 8q^{27} - 5q^{28} - 2q^{30} + 8q^{31} + q^{32} - 4q^{33} - 4q^{35} - 8q^{36} - 2q^{37} - 4q^{39} - 4q^{40} + 4q^{41} - 2q^{42} - 4q^{44} - 2q^{46} - 8q^{48} - 11q^{49} + 3q^{50} + 2q^{51} - 8q^{55} - 7q^{56} + 8q^{58} - 4q^{60} - 4q^{63} - 5q^{64} + 2q^{65} - 6q^{67} + 2q^{68} + 6q^{70} + 2q^{71} + 8q^{72} - 6q^{73} - 2q^{74} + 4q^{76} + 4q^{78} - 10q^{79} + 10q^{81} + 4q^{83} + 10q^{88} - 4q^{89} + 4q^{90} - 6q^{91} + 2q^{92} - 8q^{95} - 4q^{97} + q^{98} + 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(504))$$

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
504.1.d $$\chi_{504}(449, \cdot)$$ None 0 1
504.1.e $$\chi_{504}(251, \cdot)$$ 504.1.e.a 4 1
504.1.f $$\chi_{504}(433, \cdot)$$ None 0 1
504.1.g $$\chi_{504}(379, \cdot)$$ None 0 1
504.1.l $$\chi_{504}(181, \cdot)$$ 504.1.l.a 1 1
504.1.l.b 2
504.1.m $$\chi_{504}(127, \cdot)$$ None 0 1
504.1.n $$\chi_{504}(197, \cdot)$$ None 0 1
504.1.o $$\chi_{504}(503, \cdot)$$ None 0 1
504.1.u $$\chi_{504}(59, \cdot)$$ None 0 2
504.1.v $$\chi_{504}(65, \cdot)$$ None 0 2
504.1.ba $$\chi_{504}(67, \cdot)$$ 504.1.ba.a 4 2
504.1.bb $$\chi_{504}(313, \cdot)$$ None 0 2
504.1.bc $$\chi_{504}(143, \cdot)$$ None 0 2
504.1.bd $$\chi_{504}(53, \cdot)$$ None 0 2
504.1.bg $$\chi_{504}(29, \cdot)$$ None 0 2
504.1.bh $$\chi_{504}(383, \cdot)$$ None 0 2
504.1.bi $$\chi_{504}(149, \cdot)$$ None 0 2
504.1.bj $$\chi_{504}(167, \cdot)$$ None 0 2
504.1.bn $$\chi_{504}(13, \cdot)$$ 504.1.bn.a 2 2
504.1.bn.b 2
504.1.bn.c 4
504.1.bo $$\chi_{504}(151, \cdot)$$ None 0 2
504.1.bp $$\chi_{504}(229, \cdot)$$ None 0 2
504.1.bq $$\chi_{504}(295, \cdot)$$ None 0 2
504.1.bv $$\chi_{504}(415, \cdot)$$ None 0 2
504.1.bw $$\chi_{504}(325, \cdot)$$ 504.1.bw.a 4 2
504.1.bx $$\chi_{504}(163, \cdot)$$ None 0 2
504.1.by $$\chi_{504}(73, \cdot)$$ None 0 2
504.1.cd $$\chi_{504}(97, \cdot)$$ None 0 2
504.1.ce $$\chi_{504}(403, \cdot)$$ 504.1.ce.a 4 2
504.1.cf $$\chi_{504}(241, \cdot)$$ None 0 2
504.1.cg $$\chi_{504}(43, \cdot)$$ None 0 2
504.1.cl $$\chi_{504}(113, \cdot)$$ None 0 2
504.1.cm $$\chi_{504}(131, \cdot)$$ None 0 2
504.1.cn $$\chi_{504}(137, \cdot)$$ None 0 2
504.1.co $$\chi_{504}(83, \cdot)$$ None 0 2
504.1.ct $$\chi_{504}(395, \cdot)$$ None 0 2
504.1.cu $$\chi_{504}(233, \cdot)$$ 504.1.cu.a 4 2
504.1.cv $$\chi_{504}(79, \cdot)$$ None 0 2
504.1.cw $$\chi_{504}(61, \cdot)$$ None 0 2
504.1.da $$\chi_{504}(47, \cdot)$$ None 0 2
504.1.db $$\chi_{504}(221, \cdot)$$ None 0 2

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(504))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(504)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 2}$$