Properties

Label 68.1.f
Level 68
Weight 1
Character orbit f
Rep. character \(\chi_{68}(47,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 2
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.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 68 \)
Character field: \(\Q(i)\)
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 6 6 0
Cusp forms 2 2 0
Eisenstein series 4 4 0

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

\( 2q - 2q^{4} - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{5} + 2q^{10} + 2q^{16} - 2q^{17} - 2q^{18} + 2q^{20} + 2q^{29} - 2q^{37} - 2q^{40} + 2q^{41} + 2q^{45} - 2q^{50} - 2q^{58} - 2q^{61} - 2q^{64} + 2q^{68} + 2q^{72} + 2q^{73} + 2q^{74} - 2q^{80} - 2q^{81} + 2q^{82} + 2q^{85} + 2q^{90} - 2q^{97} + 2q^{98} + O(q^{100}) \)

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.f.a \(2\) \(0.034\) \(\Q(\sqrt{-1}) \) \(D_{4}\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}-q^{4}+(-1-i)q^{5}-iq^{8}+\cdots\)

Hecke characteristic polynomials

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