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Properties

Label 68.2.e
Level $68$
Weight $2$
Character orbit 68.e
Rep. character $\chi_{68}(13,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $4$
Newform subspaces $1$
Sturm bound $18$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(18\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(68, [\chi])\).

Total New Old
Modular forms 24 4 20
Cusp forms 12 4 8
Eisenstein series 12 0 12

Trace form

\( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + O(q^{10}) \) \( 4 q - 2 q^{3} - 4 q^{5} + 2 q^{7} + 6 q^{11} - 4 q^{13} - 28 q^{21} - 10 q^{23} + 28 q^{27} + 8 q^{29} + 10 q^{31} + 20 q^{33} - 4 q^{35} + 12 q^{37} - 24 q^{39} + 4 q^{41} + 16 q^{45} - 16 q^{47} - 30 q^{51} - 12 q^{55} - 20 q^{57} + 4 q^{61} + 34 q^{63} + 4 q^{65} + 8 q^{67} + 36 q^{69} - 10 q^{71} - 28 q^{73} - 6 q^{75} + 6 q^{79} - 32 q^{81} - 8 q^{85} - 12 q^{89} + 24 q^{91} + 12 q^{95} - 16 q^{97} - 2 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(68, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
68.2.e.a 68.e 17.c $4$ $0.543$ \(\Q(i, \sqrt{13})\) None \(0\) \(-2\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{3}+(-1-\beta _{1})q^{5}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(68, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(68, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 2}\)