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Properties

Label 68.2.i
Level $68$
Weight $2$
Character orbit 68.i
Rep. character $\chi_{68}(3,\cdot)$
Character field $\Q(\zeta_{16})$
Dimension $56$
Newform subspaces $2$
Sturm bound $18$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 68 = 2^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 68.i (of order \(16\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 68 \)
Character field: \(\Q(\zeta_{16})\)
Newform subspaces: \( 2 \)
Sturm bound: \(18\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 88 88 0
Cusp forms 56 56 0
Eisenstein series 32 32 0

Trace form

\( 56 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} + O(q^{10}) \) \( 56 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} - 8 q^{10} - 8 q^{12} - 16 q^{13} - 8 q^{14} - 16 q^{17} - 16 q^{18} - 8 q^{20} - 16 q^{21} - 8 q^{22} + 8 q^{24} - 16 q^{25} + 24 q^{26} + 40 q^{28} - 16 q^{29} + 56 q^{30} + 32 q^{32} + 56 q^{34} + 56 q^{36} - 16 q^{37} + 32 q^{38} + 56 q^{40} - 16 q^{41} + 40 q^{42} + 24 q^{44} - 16 q^{45} + 8 q^{46} - 32 q^{48} - 16 q^{49} - 16 q^{52} - 8 q^{53} - 24 q^{54} - 48 q^{56} + 64 q^{57} - 64 q^{58} - 112 q^{60} + 16 q^{61} - 64 q^{62} - 56 q^{64} + 88 q^{65} - 96 q^{66} - 96 q^{68} + 32 q^{69} - 80 q^{70} - 112 q^{72} + 88 q^{73} - 56 q^{74} - 64 q^{76} + 16 q^{77} - 112 q^{78} - 24 q^{80} + 64 q^{81} - 48 q^{82} - 8 q^{85} + 64 q^{86} + 56 q^{88} - 16 q^{89} + 72 q^{90} + 104 q^{92} - 16 q^{93} + 88 q^{94} + 144 q^{96} - 16 q^{97} + 128 q^{98} + O(q^{100}) \)

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.i.a 68.i 68.i $8$ $0.543$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{16}]$ \(q+(-\zeta_{16}+\zeta_{16}^{5})q^{2}-2\zeta_{16}^{6}q^{4}+\cdots\)
68.2.i.b 68.i 68.i $48$ $0.543$ None \(-8\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{16}]$