# 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

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

## 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 the 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

$$S_{1}^{\mathrm{old}}(\Gamma_1(612)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(68))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(204))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - T$$)($$1 + T^{2}$$)($$1 + T + T^{2}$$)($$1 + T + T^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$1 + T^{4}$$)($$1 + T^{4}$$)($$1 + T^{8}$$)($$1 + T^{8}$$)
$3$ ($$1 + T + T^{2}$$)($$1 - T + T^{2}$$)($$1 - T^{2} + T^{4}$$)
$5$ ($$( 1 - T )( 1 + T )$$)($$1 - T^{4} + T^{8}$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 + T^{4} )$$)($$( 1 - T )^{4}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)
$7$ ($$1 + T^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$1 + T^{8}$$)($$1 + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$11$ ($$1 + T^{2}$$)($$1 - T^{4} + T^{8}$$)($$1 + T^{4}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)($$1 + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$13$ ($$( 1 + T )^{2}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{4}( 1 - T + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$17$ ($$1 + T$$)($$1 + T^{4}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{2}$$)($$( 1 - T )^{4}$$)($$( 1 - T )^{4}$$)($$( 1 + T )^{4}$$)($$( 1 + T )^{8}$$)($$( 1 - T )^{8}$$)
$19$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$23$ ($$1 + T^{2}$$)($$1 - T^{4} + T^{8}$$)($$1 + T^{4}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)($$1 + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$29$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)
$31$ ($$1 + T^{2}$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$1 + T^{4}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)($$1 + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$37$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)
$41$ ($$( 1 - T )( 1 + T )$$)($$1 - T^{4} + T^{8}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)
$43$ ($$( 1 - T )( 1 + T )$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$47$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{2} )^{4}$$)($$( 1 + T^{4} )^{4}$$)($$( 1 + T^{4} )^{4}$$)
$53$ ($$( 1 + T )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{4}$$)($$( 1 - T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{8}( 1 + T^{4} )^{2}$$)($$( 1 + T )^{8}( 1 + T^{4} )^{2}$$)
$59$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$61$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)
$67$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{2} )^{4}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{8}$$)($$( 1 + T^{2} )^{8}$$)
$71$ ($$1 + T^{2}$$)($$( 1 + T^{4} )^{2}$$)($$1 + T^{4}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{2}$$)($$( 1 - T^{2} + T^{4} )^{2}$$)($$1 + T^{8}$$)($$1 + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$73$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$( 1 - T )^{2}( 1 + T^{2} )$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{2}( 1 + T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T )^{4}( 1 + T^{4} )$$)($$( 1 + T )^{4}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)($$( 1 + T^{2} )^{4}( 1 + T^{8} )$$)
$79$ ($$1 + T^{2}$$)($$( 1 + T )^{4}( 1 + T^{2} )^{2}$$)($$1 + T^{4}$$)($$( 1 - T )^{2}( 1 + T + T^{2} )$$)($$( 1 + T )^{2}( 1 - T + T^{2} )$$)($$( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} )$$)($$1 + T^{8}$$)($$1 + T^{8}$$)($$1 + T^{16}$$)($$1 + T^{16}$$)
$83$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$89$ ($$( 1 - T )^{2}$$)($$( 1 - T )^{4}( 1 + T )^{4}$$)($$( 1 + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 + T + T^{2} )^{2}$$)($$( 1 - T + T^{2} )^{4}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T^{8} )^{2}$$)($$( 1 + T^{8} )^{2}$$)
$97$ ($$( 1 - T )( 1 + T )$$)($$( 1 + T^{4} )^{2}$$)($$( 1 + T )^{2}( 1 + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )( 1 + T + T^{2} )$$)($$( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{2} )^{2}( 1 + T^{4} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)($$( 1 + T^{4} )^{2}( 1 + T^{8} )$$)