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Properties

Label 85.2.e
Level $85$
Weight $2$
Character orbit 85.e
Rep. character $\chi_{85}(21,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $1$
Sturm bound $18$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.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 \(M_{2}(85, [\chi])\).

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

Trace form

\( 12 q - 4 q^{3} - 12 q^{4} + O(q^{10}) \) \( 12 q - 4 q^{3} - 12 q^{4} - 4 q^{10} - 4 q^{11} - 8 q^{12} - 4 q^{14} + 4 q^{16} + 12 q^{17} + 28 q^{18} - 8 q^{20} - 16 q^{21} + 20 q^{22} + 12 q^{23} + 4 q^{24} - 4 q^{27} + 4 q^{28} - 12 q^{29} - 8 q^{30} - 16 q^{33} - 12 q^{34} + 16 q^{35} + 12 q^{37} + 24 q^{38} - 20 q^{39} - 8 q^{40} - 24 q^{41} + 8 q^{44} + 8 q^{45} - 24 q^{46} - 48 q^{47} - 20 q^{48} + 4 q^{50} + 32 q^{51} - 56 q^{52} + 28 q^{54} + 40 q^{56} + 36 q^{58} + 40 q^{61} + 40 q^{62} + 12 q^{63} + 28 q^{64} + 4 q^{65} - 8 q^{67} - 40 q^{68} + 28 q^{71} + 20 q^{72} - 48 q^{73} + 28 q^{74} + 4 q^{75} - 92 q^{78} - 8 q^{79} + 16 q^{80} + 28 q^{81} + 40 q^{82} - 4 q^{85} - 96 q^{86} - 72 q^{88} + 24 q^{89} - 16 q^{90} - 36 q^{91} - 16 q^{92} - 8 q^{95} - 32 q^{96} + 4 q^{97} + 44 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(85, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
85.2.e.a 85.e 17.c $12$ $0.679$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{10}q^{2}+(-\beta _{3}+\beta _{6})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)