Properties

Label 117.2.b
Level $117$
Weight $2$
Character orbit 117.b
Rep. character $\chi_{117}(64,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $2$
Sturm bound $28$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(117, [\chi])\).

Total New Old
Modular forms 18 8 10
Cusp forms 10 6 4
Eisenstein series 8 2 6

Trace form

\( 6 q - 10 q^{4} + O(q^{10}) \) \( 6 q - 10 q^{4} - 4 q^{10} - 2 q^{13} - 12 q^{14} + 26 q^{16} + 12 q^{17} - 16 q^{22} - 10 q^{25} + 12 q^{26} - 12 q^{29} + 12 q^{38} + 52 q^{40} - 24 q^{43} + 18 q^{49} - 50 q^{52} - 12 q^{53} + 32 q^{55} - 12 q^{56} - 4 q^{61} - 12 q^{62} - 74 q^{64} - 12 q^{68} + 24 q^{74} + 24 q^{77} - 16 q^{79} + 92 q^{82} + 64 q^{88} + 24 q^{91} + 8 q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(117, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
117.2.b.a 117.b 13.b $2$ $0.934$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}-q^{4}-2\zeta_{6}q^{7}-\zeta_{6}q^{8}+2\zeta_{6}q^{11}+\cdots\)
117.2.b.b 117.b 13.b $4$ $0.934$ 4.0.8112.1 \(\Q(\sqrt{-39}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{2}q^{2}+(-2-\beta _{3})q^{4}-\beta _{1}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(117, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)