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

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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

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