## Defining parameters

 Level: $$N$$ = $$59$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$290$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(59))$$.

Total New Old
Modular forms 30 30 0
Cusp forms 1 1 0
Eisenstein series 29 29 0

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 1 0 0 0

## Trace form

 $$q - q^{3} + q^{4} - q^{5} - q^{7} + O(q^{10})$$ $$q - q^{3} + q^{4} - q^{5} - q^{7} - q^{12} + q^{15} + q^{16} + 2q^{17} - q^{19} - q^{20} + q^{21} + q^{27} - q^{28} - q^{29} + q^{35} - q^{41} - q^{48} - 2q^{51} - q^{53} + q^{57} + q^{59} + q^{60} + q^{64} + 2q^{68} + 2q^{71} - q^{76} - q^{79} - q^{80} - q^{81} + q^{84} - 2q^{85} + q^{87} + q^{95} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(59))$$

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
59.1.b $$\chi_{59}(58, \cdot)$$ 59.1.b.a 1 1
59.1.d $$\chi_{59}(2, \cdot)$$ None 0 28

## Hecke characteristic polynomials

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