Properties

Label 85.2.j
Level $85$
Weight $2$
Character orbit 85.j
Rep. character $\chi_{85}(4,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $16$
Newform subspaces $3$
Sturm bound $18$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 85 = 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 85.j (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 85 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(85, [\chi])\).

Total New Old
Modular forms 24 24 0
Cusp forms 16 16 0
Eisenstein series 8 8 0

Trace form

\( 16 q + 8 q^{4} - 6 q^{5} - 16 q^{6} + O(q^{10}) \) \( 16 q + 8 q^{4} - 6 q^{5} - 16 q^{6} - 2 q^{10} + 4 q^{11} - 4 q^{14} - 26 q^{20} - 44 q^{24} - 8 q^{29} + 40 q^{30} + 4 q^{34} + 32 q^{35} + 12 q^{39} + 18 q^{40} + 4 q^{41} + 40 q^{44} - 14 q^{45} - 8 q^{46} - 52 q^{50} + 32 q^{51} + 116 q^{54} + 24 q^{55} - 32 q^{56} - 44 q^{61} - 48 q^{64} + 8 q^{65} - 96 q^{69} - 28 q^{71} + 16 q^{74} + 52 q^{75} - 32 q^{79} + 18 q^{80} - 40 q^{81} + 16 q^{84} - 58 q^{85} + 32 q^{86} + 8 q^{89} - 82 q^{90} + 36 q^{91} - 36 q^{95} + 16 q^{96} + 24 q^{99} + O(q^{100}) \)

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.j.a 85.j 85.j $2$ $0.679$ \(\Q(\sqrt{-1}) \) None \(-2\) \(-2\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}+(-1+i)q^{3}-q^{4}+(-2-i)q^{5}+\cdots\)
85.2.j.b 85.j 85.j $2$ $0.679$ \(\Q(\sqrt{-1}) \) None \(2\) \(2\) \(-2\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+(1-i)q^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
85.2.j.c 85.j 85.j $12$ $0.679$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}+(-\beta _{2}-\beta _{5}+\beta _{7})q^{3}+(1+\cdots)q^{4}+\cdots\)