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
Newforms 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: \(\Q(i)\)
Newforms: \( 1 \)
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

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

Decomposition of \(S_{1}^{\mathrm{new}}(68, [\chi])\) into irreducible Hecke orbits

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 \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}-q^{4}+(-1-i)q^{5}-iq^{8}+\cdots\)