Properties

Label 117.2.q
Level $117$
Weight $2$
Character orbit 117.q
Rep. character $\chi_{117}(10,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $10$
Newform subspaces $4$
Sturm bound $28$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.q (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(28\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

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

Total New Old
Modular forms 36 14 22
Cusp forms 20 10 10
Eisenstein series 16 4 12

Trace form

\( 10 q + 3 q^{2} + 3 q^{4} - 6 q^{7} + O(q^{10}) \) \( 10 q + 3 q^{2} + 3 q^{4} - 6 q^{7} + 7 q^{10} + 6 q^{11} + 7 q^{13} - q^{16} - 3 q^{17} - 18 q^{19} - 15 q^{20} - 20 q^{22} - 12 q^{23} - 3 q^{26} - 48 q^{28} + 9 q^{29} + 9 q^{32} + 6 q^{35} + 9 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} + 2 q^{43} + 42 q^{46} + 19 q^{49} + 6 q^{50} - 6 q^{52} - 18 q^{53} - 8 q^{55} - 21 q^{58} - 6 q^{59} - q^{61} - 6 q^{62} + 82 q^{64} + 3 q^{65} - 12 q^{67} + 3 q^{68} + 12 q^{71} + 15 q^{74} - 42 q^{76} - 12 q^{77} + 44 q^{79} + 39 q^{80} + 19 q^{82} + 39 q^{85} + 20 q^{88} - 66 q^{91} + 12 q^{92} - 26 q^{94} - 18 q^{95} - 6 q^{97} - 21 q^{98} + O(q^{100}) \)

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.q.a 117.q 13.e $2$ $0.934$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{6}]$ \(q-2\zeta_{6}q^{4}+(2-4\zeta_{6})q^{5}+(-2+\zeta_{6})q^{7}+\cdots\)
117.2.q.b 117.q 13.e $2$ $0.934$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(9\) $\mathrm{U}(1)[D_{6}]$ \(q-2\zeta_{6}q^{4}+(6-3\zeta_{6})q^{7}+(-4+3\zeta_{6})q^{13}+\cdots\)
117.2.q.c 117.q 13.e $2$ $0.934$ \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+(-1+2\zeta_{6})q^{5}+\cdots\)
117.2.q.d 117.q 13.e $4$ $0.934$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}-\beta _{3}q^{5}+(-4+2\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(117, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)