Properties

Label 68.2.h
Level $68$
Weight $2$
Character orbit 68.h
Rep. character $\chi_{68}(9,\cdot)$
Character field $\Q(\zeta_{8})$
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.h (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(\zeta_{8})\)
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 48 4 44
Cusp forms 24 4 20
Eisenstein series 24 0 24

Trace form

\( 4 q + 4 q^{5} - 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{27} + 12 q^{29} + 24 q^{31} + 8 q^{33} + 4 q^{37} + 24 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} - 4 q^{49} - 8 q^{51} - 28 q^{53} - 4 q^{57} - 28 q^{59} + 4 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69} + 16 q^{71} - 12 q^{73} + 8 q^{75} - 4 q^{77} - 8 q^{79} + 12 q^{83} - 20 q^{85} - 4 q^{87} + 8 q^{91} + 28 q^{93} + 12 q^{95} + 12 q^{97} + 12 q^{99} + O(q^{100}) \)

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.h.a 68.h 17.d $4$ $0.543$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+(\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(1-\zeta_{8})q^{5}+(\zeta_{8}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(68, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 2}\)