Properties

Label 112.2.f
Level 112
Weight 2
Character orbit f
Rep. character \(\chi_{112}(111,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 2
Sturm bound 32
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(112, [\chi])\).

Total New Old
Modular forms 22 4 18
Cusp forms 10 4 6
Eisenstein series 12 0 12

Trace form

\( 4q + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{9} - 16q^{21} - 28q^{25} + 24q^{29} - 8q^{37} + 4q^{49} + 24q^{53} + 16q^{57} + 48q^{65} - 24q^{77} - 44q^{81} - 64q^{93} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(112, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
112.2.f.a \(2\) \(0.894\) \(\Q(\sqrt{-3}) \) None \(0\) \(-4\) \(0\) \(4\) \(q-2q^{3}-2\zeta_{6}q^{5}+(2-\zeta_{6})q^{7}+q^{9}+\cdots\)
112.2.f.b \(2\) \(0.894\) \(\Q(\sqrt{-3}) \) None \(0\) \(4\) \(0\) \(-4\) \(q+2q^{3}+2\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(112, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(112, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( ( 1 + 2 T + 3 T^{2} )^{2} \))(\( ( 1 - 2 T + 3 T^{2} )^{2} \))
$5$ (\( 1 + 2 T^{2} + 25 T^{4} \))(\( 1 + 2 T^{2} + 25 T^{4} \))
$7$ (\( 1 - 4 T + 7 T^{2} \))(\( 1 + 4 T + 7 T^{2} \))
$11$ (\( 1 - 10 T^{2} + 121 T^{4} \))(\( 1 - 10 T^{2} + 121 T^{4} \))
$13$ (\( 1 - 14 T^{2} + 169 T^{4} \))(\( 1 - 14 T^{2} + 169 T^{4} \))
$17$ (\( ( 1 - 17 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} )^{2} \))
$19$ (\( ( 1 + 2 T + 19 T^{2} )^{2} \))(\( ( 1 - 2 T + 19 T^{2} )^{2} \))
$23$ (\( 1 - 34 T^{2} + 529 T^{4} \))(\( 1 - 34 T^{2} + 529 T^{4} \))
$29$ (\( ( 1 - 6 T + 29 T^{2} )^{2} \))(\( ( 1 - 6 T + 29 T^{2} )^{2} \))
$31$ (\( ( 1 - 8 T + 31 T^{2} )^{2} \))(\( ( 1 + 8 T + 31 T^{2} )^{2} \))
$37$ (\( ( 1 + 2 T + 37 T^{2} )^{2} \))(\( ( 1 + 2 T + 37 T^{2} )^{2} \))
$41$ (\( 1 - 34 T^{2} + 1681 T^{4} \))(\( 1 - 34 T^{2} + 1681 T^{4} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))(\( ( 1 - 8 T + 43 T^{2} )( 1 + 8 T + 43 T^{2} ) \))
$47$ (\( ( 1 + 47 T^{2} )^{2} \))(\( ( 1 + 47 T^{2} )^{2} \))
$53$ (\( ( 1 - 6 T + 53 T^{2} )^{2} \))(\( ( 1 - 6 T + 53 T^{2} )^{2} \))
$59$ (\( ( 1 + 6 T + 59 T^{2} )^{2} \))(\( ( 1 - 6 T + 59 T^{2} )^{2} \))
$61$ (\( 1 - 110 T^{2} + 3721 T^{4} \))(\( 1 - 110 T^{2} + 3721 T^{4} \))
$67$ (\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))(\( ( 1 - 16 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} ) \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 130 T^{2} + 5041 T^{4} \))
$73$ (\( 1 - 98 T^{2} + 5329 T^{4} \))(\( 1 - 98 T^{2} + 5329 T^{4} \))
$79$ (\( 1 - 146 T^{2} + 6241 T^{4} \))(\( 1 - 146 T^{2} + 6241 T^{4} \))
$83$ (\( ( 1 - 6 T + 83 T^{2} )^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 130 T^{2} + 7921 T^{4} \))(\( 1 - 130 T^{2} + 7921 T^{4} \))
$97$ (\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))(\( ( 1 - 14 T + 97 T^{2} )( 1 + 14 T + 97 T^{2} ) \))
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