# Properties

 Label 57.1.h Level 57 Weight 1 Character orbit h Rep. character $$\chi_{57}(11,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 2 Newform subspaces 1 Sturm bound 6 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$57 = 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 57.h (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$57$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$6$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(57, [\chi])$$.

Total New Old
Modular forms 6 6 0
Cusp forms 2 2 0
Eisenstein series 4 4 0

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

 $$2q - q^{3} - q^{4} - 2q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{4} - 2q^{7} - q^{9} + 2q^{12} + q^{13} - q^{16} + 2q^{19} + q^{21} - q^{25} + 2q^{27} + q^{28} - 2q^{31} - q^{36} - 2q^{37} - 2q^{39} + q^{43} - q^{48} + q^{52} - q^{57} + q^{61} + q^{63} + 2q^{64} + q^{67} + q^{73} + 2q^{75} - q^{76} + q^{79} - q^{81} - 2q^{84} - q^{91} + q^{93} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(57, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field Image CM RM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
57.1.h.a $$2$$ $$0.028$$ $$\Q(\sqrt{-3})$$ $$D_{3}$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$0$$ $$-2$$ $$q+\zeta_{6}^{2}q^{3}-\zeta_{6}q^{4}-q^{7}-\zeta_{6}q^{9}+q^{12}+\cdots$$

## Hecke characteristic polynomials

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