# Properties

 Label 612.1 Level 612 Weight 1 Dimension 39 Nonzero newspaces 6 Newform subspaces 10 Sturm bound 20736 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$612 = 2^{2} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$20736$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(612))$$.

Total New Old
Modular forms 692 173 519
Cusp forms 52 39 13
Eisenstein series 640 134 506

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 35 0 4 0

## Trace form

 $$39 q + q^{2} - 5 q^{4} + 2 q^{5} + q^{8} + O(q^{10})$$ $$39 q + q^{2} - 5 q^{4} + 2 q^{5} + q^{8} - 6 q^{10} + 2 q^{13} - 9 q^{16} + 9 q^{17} - 4 q^{18} - 2 q^{20} + 8 q^{21} - 11 q^{25} - 10 q^{26} - 2 q^{29} - 4 q^{31} + q^{32} - 4 q^{33} - q^{34} - 2 q^{37} - 2 q^{40} - 2 q^{41} - 4 q^{42} - 5 q^{49} + 3 q^{50} - 2 q^{52} - 10 q^{53} - 4 q^{55} - 2 q^{58} - 2 q^{61} + 7 q^{64} + 8 q^{66} - 7 q^{68} + 8 q^{69} - 4 q^{72} - 10 q^{73} - 2 q^{74} + 4 q^{77} - 4 q^{79} + 2 q^{80} - 4 q^{81} - 6 q^{82} - 4 q^{84} - 10 q^{85} + 2 q^{89} - 4 q^{93} - 2 q^{97} + 13 q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(612))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
612.1.d $$\chi_{612}(341, \cdot)$$ None 0 1
612.1.e $$\chi_{612}(271, \cdot)$$ 612.1.e.a 1 1
612.1.f $$\chi_{612}(307, \cdot)$$ None 0 1
612.1.g $$\chi_{612}(305, \cdot)$$ None 0 1
612.1.j $$\chi_{612}(89, \cdot)$$ 612.1.j.a 4 2
612.1.l $$\chi_{612}(55, \cdot)$$ 612.1.l.a 2 2
612.1.o $$\chi_{612}(101, \cdot)$$ None 0 2
612.1.p $$\chi_{612}(103, \cdot)$$ None 0 2
612.1.q $$\chi_{612}(67, \cdot)$$ 612.1.q.a 2 2
612.1.q.b 2
612.1.q.c 4
612.1.r $$\chi_{612}(137, \cdot)$$ None 0 2
612.1.u $$\chi_{612}(53, \cdot)$$ None 0 4
612.1.v $$\chi_{612}(19, \cdot)$$ 612.1.v.a 4 4
612.1.v.b 4
612.1.y $$\chi_{612}(149, \cdot)$$ None 0 4
612.1.ba $$\chi_{612}(115, \cdot)$$ None 0 4
612.1.be $$\chi_{612}(37, \cdot)$$ None 0 8
612.1.bf $$\chi_{612}(71, \cdot)$$ 612.1.bf.a 8 8
612.1.bf.b 8
612.1.bg $$\chi_{612}(77, \cdot)$$ None 0 8
612.1.bh $$\chi_{612}(43, \cdot)$$ None 0 8
612.1.bm $$\chi_{612}(11, \cdot)$$ None 0 16
612.1.bn $$\chi_{612}(61, \cdot)$$ None 0 16

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(612))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(612)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 12}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 9}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(68))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(153))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(204))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(306))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(612))$$$$^{\oplus 1}$$