# Properties

 Label 56.2.e Level 56 Weight 2 Character orbit e Rep. character $$\chi_{56}(27,\cdot)$$ Character field $$\Q$$ Dimension 6 Newform subspaces 2 Sturm bound 16 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$56 = 2^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 56.e (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$56$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$16$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(56, [\chi])$$.

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

## Trace form

 $$6q - 3q^{2} - 3q^{4} + 3q^{8} - 6q^{9} + O(q^{10})$$ $$6q - 3q^{2} - 3q^{4} + 3q^{8} - 6q^{9} + 3q^{14} - 15q^{16} + 15q^{18} - 12q^{22} - 6q^{25} + 15q^{28} + 24q^{30} + 27q^{32} - 24q^{35} - 9q^{36} - 24q^{42} + 12q^{44} - 30q^{46} + 6q^{49} - 9q^{50} + 48q^{51} - 15q^{56} + 24q^{57} - 12q^{58} - 48q^{60} + 9q^{64} - 24q^{65} + 24q^{70} - 39q^{72} + 60q^{74} - 24q^{78} - 18q^{81} + 48q^{84} + 36q^{86} + 36q^{88} + 24q^{91} + 18q^{92} - 27q^{98} - 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(56, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
56.2.e.a $$2$$ $$0.447$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ $$1$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{2}+(-2+\beta )q^{4}+(1-2\beta )q^{7}+\cdots$$
56.2.e.b $$4$$ $$0.447$$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$0$$ $$q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+\cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$1 - T + 2 T^{2}$$)($$( 1 + 2 T + 2 T^{2} )^{2}$$)
$3$ ($$( 1 - 3 T^{2} )^{2}$$)($$( 1 + 9 T^{4} )^{2}$$)
$5$ ($$( 1 + 5 T^{2} )^{2}$$)($$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$)
$7$ ($$1 + 7 T^{2}$$)($$1 - 10 T^{2} + 49 T^{4}$$)
$11$ ($$( 1 + 4 T + 11 T^{2} )^{2}$$)($$( 1 - 2 T + 11 T^{2} )^{4}$$)
$13$ ($$( 1 + 13 T^{2} )^{2}$$)($$( 1 + 20 T^{2} + 169 T^{4} )^{2}$$)
$17$ ($$( 1 - 17 T^{2} )^{2}$$)($$( 1 - 10 T^{2} + 289 T^{4} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} )^{2}$$)($$( 1 - 32 T^{2} + 361 T^{4} )^{2}$$)
$23$ ($$( 1 - 8 T + 23 T^{2} )( 1 + 8 T + 23 T^{2} )$$)($$( 1 - 30 T^{2} + 529 T^{4} )^{2}$$)
$29$ ($$( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} )$$)($$( 1 - 10 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$)
$31$ ($$( 1 + 31 T^{2} )^{2}$$)($$( 1 + 38 T^{2} + 961 T^{4} )^{2}$$)
$37$ ($$( 1 - 6 T + 37 T^{2} )( 1 + 6 T + 37 T^{2} )$$)($$( 1 - 10 T^{2} + 1369 T^{4} )^{2}$$)
$41$ ($$( 1 - 41 T^{2} )^{2}$$)($$( 1 - 41 T^{2} )^{4}$$)
$43$ ($$( 1 - 12 T + 43 T^{2} )^{2}$$)($$( 1 + 6 T + 43 T^{2} )^{4}$$)
$47$ ($$( 1 + 47 T^{2} )^{2}$$)($$( 1 + 70 T^{2} + 2209 T^{4} )^{2}$$)
$53$ ($$( 1 - 10 T + 53 T^{2} )( 1 + 10 T + 53 T^{2} )$$)($$( 1 - 14 T + 53 T^{2} )^{2}( 1 + 14 T + 53 T^{2} )^{2}$$)
$59$ ($$( 1 - 59 T^{2} )^{2}$$)($$( 1 - 112 T^{2} + 3481 T^{4} )^{2}$$)
$61$ ($$( 1 + 61 T^{2} )^{2}$$)($$( 1 + 68 T^{2} + 3721 T^{4} )^{2}$$)
$67$ ($$( 1 - 4 T + 67 T^{2} )^{2}$$)($$( 1 + 2 T + 67 T^{2} )^{4}$$)
$71$ ($$( 1 - 16 T + 71 T^{2} )( 1 + 16 T + 71 T^{2} )$$)($$( 1 - 42 T^{2} + 5041 T^{4} )^{2}$$)
$73$ ($$( 1 - 73 T^{2} )^{2}$$)($$( 1 + 70 T^{2} + 5329 T^{4} )^{2}$$)
$79$ ($$( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} )$$)($$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$)
$83$ ($$( 1 - 83 T^{2} )^{2}$$)($$( 1 - 160 T^{2} + 6889 T^{4} )^{2}$$)
$89$ ($$( 1 - 89 T^{2} )^{2}$$)($$( 1 + 38 T^{2} + 7921 T^{4} )^{2}$$)
$97$ ($$( 1 - 97 T^{2} )^{2}$$)($$( 1 - 170 T^{2} + 9409 T^{4} )^{2}$$)