## Defining parameters

 Level: $$N$$ = $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$192$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(56))$$.

Total New Old
Modular forms 37 11 26
Cusp forms 1 1 0
Eisenstein series 36 10 26

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

 $$q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + O(q^{10})$$ $$q - q^{2} + q^{4} - q^{7} - q^{8} - q^{9} + q^{14} + q^{16} + q^{18} + 2q^{23} - q^{25} - q^{28} - q^{32} - q^{36} - 2q^{46} + q^{49} + q^{50} + q^{56} + q^{63} + q^{64} - 2q^{71} + q^{72} - 2q^{79} + q^{81} + 2q^{92} - q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$

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
56.1.c $$\chi_{56}(41, \cdot)$$ None 0 1
56.1.d $$\chi_{56}(15, \cdot)$$ None 0 1
56.1.g $$\chi_{56}(43, \cdot)$$ None 0 1
56.1.h $$\chi_{56}(13, \cdot)$$ 56.1.h.a 1 1
56.1.j $$\chi_{56}(5, \cdot)$$ None 0 2
56.1.k $$\chi_{56}(11, \cdot)$$ None 0 2
56.1.n $$\chi_{56}(23, \cdot)$$ None 0 2
56.1.o $$\chi_{56}(17, \cdot)$$ None 0 2

## Hecke characteristic polynomials

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