Properties

Label 56.2.b
Level 56
Weight 2
Character orbit b
Rep. character \(\chi_{56}(29,\cdot)\)
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.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 8 \)
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 6 4
Cusp forms 6 6 0
Eisenstein series 4 0 4

Trace form

\( 6q - q^{2} - 3q^{4} + 2q^{6} - 2q^{7} - 7q^{8} - 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 3q^{4} + 2q^{6} - 2q^{7} - 7q^{8} - 6q^{9} - 4q^{10} + 14q^{12} + q^{14} + 8q^{15} + q^{16} - 4q^{17} - 15q^{18} + 4q^{20} - 6q^{22} - 8q^{23} + 6q^{24} - 2q^{25} + 20q^{26} - 5q^{28} + 16q^{30} - 16q^{31} + 9q^{32} + 8q^{33} - 2q^{34} + 11q^{36} + 18q^{38} + 8q^{39} - 28q^{40} - 4q^{41} - 10q^{42} - 18q^{44} + 16q^{46} - 10q^{48} + 6q^{49} + 19q^{50} - 4q^{52} - 44q^{54} + 32q^{55} + 7q^{56} - 8q^{57} - 20q^{58} - 24q^{60} - 32q^{62} + 10q^{63} - 15q^{64} + 16q^{65} - 4q^{66} + 26q^{68} + 12q^{70} - 32q^{71} + 31q^{72} - 20q^{73} - 12q^{74} + 14q^{76} - 16q^{78} + 16q^{79} + 36q^{80} + 14q^{81} + 38q^{82} - 14q^{84} + 26q^{86} + 32q^{87} + 38q^{88} - 20q^{89} + 20q^{90} + 8q^{92} - 8q^{95} - 58q^{96} - 4q^{97} - q^{98} + O(q^{100}) \)

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.b.a \(2\) \(0.447\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(2\) \(q+\beta q^{2}+\beta q^{3}-2q^{4}-\beta q^{5}-2q^{6}+\cdots\)
56.2.b.b \(4\) \(0.447\) 4.0.2312.1 None \(-1\) \(0\) \(0\) \(-4\) \(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))
$3$ (\( 1 - 4 T^{2} + 9 T^{4} \))(\( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \))
$5$ (\( 1 - 8 T^{2} + 25 T^{4} \))(\( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \))
$7$ (\( ( 1 - T )^{2} \))(\( ( 1 + T )^{4} \))
$11$ (\( ( 1 - 6 T + 11 T^{2} )( 1 + 6 T + 11 T^{2} ) \))(\( 1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8} \))
$13$ (\( 1 - 8 T^{2} + 169 T^{4} \))(\( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} \))
$17$ (\( ( 1 + 6 T + 17 T^{2} )^{2} \))(\( ( 1 - 2 T + 17 T^{2} )^{4} \))
$19$ (\( 1 - 20 T^{2} + 361 T^{4} \))(\( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} \))
$23$ (\( ( 1 + 6 T + 23 T^{2} )^{2} \))(\( ( 1 - 2 T + 30 T^{2} - 46 T^{3} + 529 T^{4} )^{2} \))
$29$ (\( 1 - 50 T^{2} + 841 T^{4} \))(\( 1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8} \))
$31$ (\( ( 1 + 4 T + 31 T^{2} )^{2} \))(\( ( 1 + 4 T - 2 T^{2} + 124 T^{3} + 961 T^{4} )^{2} \))
$37$ (\( 1 - 2 T^{2} + 1369 T^{4} \))(\( 1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8} \))
$41$ (\( ( 1 - 6 T + 41 T^{2} )^{2} \))(\( ( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 10 T + 43 T^{2} )( 1 + 10 T + 43 T^{2} ) \))(\( 1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8} \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{4} \))
$53$ (\( 1 - 74 T^{2} + 2809 T^{4} \))(\( 1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8} \))
$59$ (\( 1 - 116 T^{2} + 3481 T^{4} \))(\( 1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8} \))
$61$ (\( 1 + 40 T^{2} + 3721 T^{4} \))(\( 1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8} \))
$67$ (\( ( 1 - 67 T^{2} )^{2} \))(\( 1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8} \))
$71$ (\( ( 1 + 71 T^{2} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 - 2 T + 73 T^{2} )^{2} \))(\( ( 1 + 6 T + 73 T^{2} )^{4} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )^{2} \))(\( ( 1 + 79 T^{2} )^{4} \))
$83$ (\( 1 + 76 T^{2} + 6889 T^{4} \))(\( 1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8} \))
$89$ (\( ( 1 - 6 T + 89 T^{2} )^{2} \))(\( ( 1 + 16 T + 174 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 + 10 T + 97 T^{2} )^{2} \))(\( ( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \))
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