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Properties

Label 117.1.j
Level 117
Weight 1
Character orbit j
Rep. character \(\chi_{117}(73,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 2
Newform subspaces 1
Sturm bound 14
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 117.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(14\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(117, [\chi])\).

Total New Old
Modular forms 10 4 6
Cusp forms 2 2 0
Eisenstein series 8 2 6

The following table gives the dimensions of subspaces with specified projective image type.

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 2 0 0 0

Trace form

\( 2q - 2q^{7} + O(q^{10}) \) \( 2q - 2q^{7} - 2q^{16} - 2q^{19} + 2q^{28} + 2q^{31} + 2q^{37} + 2q^{52} - 2q^{67} - 2q^{73} - 2q^{76} - 2q^{91} + 2q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{1}^{\mathrm{new}}(117, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
117.1.j.a \(2\) \(0.058\) \(\Q(\sqrt{-1}) \) \(D_{4}\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-2\) \(q+iq^{4}+(-1-i)q^{7}-iq^{13}-q^{16}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ 1
$5$ \( 1 + T^{4} \)
$7$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$11$ \( 1 + T^{4} \)
$13$ \( 1 + T^{2} \)
$17$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$19$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$23$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$29$ \( ( 1 + T^{2} )^{2} \)
$31$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
$37$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
$41$ \( 1 + T^{4} \)
$43$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
$47$ \( 1 + T^{4} \)
$53$ \( ( 1 + T^{2} )^{2} \)
$59$ \( 1 + T^{4} \)
$61$ \( ( 1 + T^{2} )^{2} \)
$67$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$71$ \( 1 + T^{4} \)
$73$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
$79$ \( ( 1 + T^{2} )^{2} \)
$83$ \( 1 + T^{4} \)
$89$ \( 1 + T^{4} \)
$97$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
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