Properties

Label 68.1.d
Level 68
Weight 1
Character orbit d
Rep. character \(\chi_{68}(67,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 68.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 68 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(68, [\chi])\).

Total New Old
Modular forms 3 3 0
Cusp forms 1 1 0
Eisenstein series 2 2 0

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 1 0 0 0

Trace form

\( q - q^{2} + q^{4} - q^{8} - q^{9} + O(q^{10}) \) \( q - q^{2} + q^{4} - q^{8} - q^{9} - 2q^{13} + q^{16} + q^{17} + q^{18} + q^{25} + 2q^{26} - q^{32} - q^{34} - q^{36} - q^{49} - q^{50} - 2q^{52} + 2q^{53} + q^{64} + q^{68} + q^{72} + q^{81} - 2q^{89} + q^{98} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(68, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
68.1.d.a \(1\) \(0.034\) \(\Q\) \(D_{2}\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-17}) \) \(\Q(\sqrt{17}) \) \(-1\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}-q^{8}-q^{9}-2q^{13}+q^{16}+\cdots\)

Hecke characteristic polynomials

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