# Properties

 Label 120.1 Level 120 Weight 1 Dimension 2 Nonzero newspaces 1 Newform subspaces 1 Sturm bound 768 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$120\( 120 = 2^{3} \cdot 3 \cdot 5$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$768$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(120))$$.

Total New Old
Modular forms 98 14 84
Cusp forms 2 2 0
Eisenstein series 96 12 84

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 2 0 0 0

## Trace form

 $$2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} - 2q^{9} + 2q^{10} + 2q^{15} + 2q^{16} + 2q^{24} - 2q^{25} - 4q^{31} + 2q^{36} - 2q^{40} + 2q^{49} + 2q^{54} - 2q^{60} - 2q^{64} + 4q^{79} + 2q^{81} - 2q^{90} - 2q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(120))$$

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
120.1.c $$\chi_{120}(89, \cdot)$$ None 0 1
120.1.e $$\chi_{120}(31, \cdot)$$ None 0 1
120.1.g $$\chi_{120}(91, \cdot)$$ None 0 1
120.1.i $$\chi_{120}(29, \cdot)$$ 120.1.i.a 2 1
120.1.j $$\chi_{120}(79, \cdot)$$ None 0 1
120.1.l $$\chi_{120}(41, \cdot)$$ None 0 1
120.1.n $$\chi_{120}(101, \cdot)$$ None 0 1
120.1.p $$\chi_{120}(19, \cdot)$$ None 0 1
120.1.q $$\chi_{120}(83, \cdot)$$ None 0 2
120.1.t $$\chi_{120}(13, \cdot)$$ None 0 2
120.1.u $$\chi_{120}(73, \cdot)$$ None 0 2
120.1.x $$\chi_{120}(23, \cdot)$$ None 0 2

## Hecke characteristic polynomials

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