Properties

Label 56.2.i
Level 56
Weight 2
Character orbit i
Rep. character \(\chi_{56}(9,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 4
Newform subspaces 2
Sturm bound 16
Trace bound 3

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Defining parameters

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

Dimensions

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

Total New Old
Modular forms 24 4 20
Cusp forms 8 4 4
Eisenstein series 16 0 16

Trace form

\( 4q - 2q^{3} + 2q^{5} - 4q^{9} + O(q^{10}) \) \( 4q - 2q^{3} + 2q^{5} - 4q^{9} - 2q^{11} - 8q^{13} - 4q^{15} + 2q^{17} - 6q^{19} - 2q^{21} + 10q^{23} + 8q^{25} + 28q^{27} - 8q^{29} + 6q^{31} + 6q^{33} + 6q^{35} + 2q^{37} - 12q^{39} - 24q^{41} - 16q^{43} + 4q^{45} - 6q^{47} + 4q^{49} - 14q^{51} + 10q^{53} - 4q^{55} + 28q^{57} - 18q^{59} + 2q^{61} - 28q^{63} - 4q^{65} - 2q^{67} - 4q^{69} + 32q^{71} - 14q^{73} + 8q^{75} + 14q^{77} + 10q^{79} - 10q^{81} + 16q^{83} + 4q^{85} + 20q^{87} + 2q^{89} + 32q^{91} - 2q^{93} + 6q^{95} + 8q^{97} - 24q^{99} + O(q^{100}) \)

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.i.a \(2\) \(0.447\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(1\) \(4\) \(q-3\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
56.2.i.b \(2\) \(0.447\) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-4\) \(q+\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(56, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(56, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))
$3$ (\( ( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \))(\( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \))
$5$ (\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))(\( 1 - T - 4 T^{2} - 5 T^{3} + 25 T^{4} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))
$11$ (\( 1 - T - 10 T^{2} - 11 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )^{2} \))(\( ( 1 + 6 T + 13 T^{2} )^{2} \))
$17$ (\( 1 + 3 T - 8 T^{2} + 51 T^{3} + 289 T^{4} \))(\( 1 - 5 T + 8 T^{2} - 85 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( ( 1 - 7 T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))
$23$ (\( 1 - 3 T - 14 T^{2} - 69 T^{3} + 529 T^{4} \))(\( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \))
$29$ (\( ( 1 + 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 2 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))(\( 1 - 5 T - 6 T^{2} - 155 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} \))(\( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} \))
$41$ (\( ( 1 + 10 T + 41 T^{2} )^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))
$43$ (\( ( 1 + 4 T + 43 T^{2} )^{2} \))(\( ( 1 + 4 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + T - 46 T^{2} + 47 T^{3} + 2209 T^{4} \))(\( 1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} \))
$59$ (\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))(\( 1 + 15 T + 166 T^{2} + 885 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 3 T - 52 T^{2} + 183 T^{3} + 3721 T^{4} \))(\( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} \))
$67$ (\( ( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( 1 - 9 T + 14 T^{2} - 603 T^{3} + 4489 T^{4} \))
$71$ (\( ( 1 - 16 T + 71 T^{2} )^{2} \))(\( ( 1 + 71 T^{2} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))
$79$ (\( 1 - 11 T + 42 T^{2} - 869 T^{3} + 6241 T^{4} \))(\( 1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4} \))
$83$ (\( ( 1 + 4 T + 83 T^{2} )^{2} \))(\( ( 1 - 12 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} \))(\( 1 + 7 T - 40 T^{2} + 623 T^{3} + 7921 T^{4} \))
$97$ (\( ( 1 - 6 T + 97 T^{2} )^{2} \))(\( ( 1 + 2 T + 97 T^{2} )^{2} \))
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