Properties

Label 117.2.a
Level $117$
Weight $2$
Character orbit 117.a
Rep. character $\chi_{117}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $3$
Sturm bound $28$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(28\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(117))\).

Total New Old
Modular forms 18 5 13
Cusp forms 11 5 6
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(13\)FrickeDim
\(+\)\(-\)$-$\(2\)
\(-\)\(+\)$-$\(2\)
\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5 q + q^{2} + 3 q^{4} - 2 q^{5} + 9 q^{8} + O(q^{10}) \) \( 5 q + q^{2} + 3 q^{4} - 2 q^{5} + 9 q^{8} - 6 q^{10} + q^{13} - 4 q^{14} - 5 q^{16} - 6 q^{17} + 4 q^{19} - 14 q^{20} - 4 q^{22} + 8 q^{23} - 5 q^{25} - 3 q^{26} - 8 q^{28} + 6 q^{29} - 11 q^{32} - 10 q^{34} + 24 q^{35} - 2 q^{37} + 8 q^{38} - 14 q^{40} - 22 q^{41} + 12 q^{43} + 8 q^{44} + 32 q^{46} + 12 q^{47} + 5 q^{49} + 7 q^{50} - q^{52} - 2 q^{53} + 8 q^{55} - 20 q^{56} + 10 q^{58} - 16 q^{59} - 18 q^{61} - 20 q^{62} - 5 q^{64} - 2 q^{65} + 28 q^{67} + 30 q^{68} + 8 q^{70} - 4 q^{71} - 6 q^{73} + 14 q^{74} + 20 q^{76} + 16 q^{77} + 2 q^{80} + 6 q^{82} - 28 q^{85} + 36 q^{86} + 12 q^{88} - 22 q^{89} + 8 q^{92} - 40 q^{94} - 16 q^{95} - 14 q^{97} - 7 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(117))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
117.2.a.a 117.a 1.a $1$ $0.934$ \(\Q\) None \(-1\) \(0\) \(-2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-2q^{5}-4q^{7}+3q^{8}+2q^{10}+\cdots\)
117.2.a.b 117.a 1.a $2$ $0.934$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+2q^{7}-\beta q^{8}-2\beta q^{11}+\cdots\)
117.2.a.c 117.a 1.a $2$ $0.934$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-2\beta q^{5}-2\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(117))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(117)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)