Embedded newform 117.2.x.a.11.1

• Label: 117.2.x.a.11.1
• Level: $117$
• Weight: $2$
• Character: 117.11
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.x (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	117.11
Dual form			117.2.x.a.32.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86694 + 1.86694i) q^{2} +(0.154854 + 1.72511i) q^{3} -4.97091i q^{4} +(-2.60400 - 0.697740i) q^{5} +(-3.50978 - 2.93158i) q^{6} +(-2.18636 - 0.585835i) q^{7} +(5.54651 + 5.54651i) q^{8} +(-2.95204 + 0.534283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} - 2 q^{3} - 6 q^{5} - 8 q^{6} - 4 q^{7} + 30 q^{8} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86694	+	1.86694i	−1.32012	+	1.32012i	−0.406452	+	0.913672i	\(0.633234\pi\)
							−0.913672	+	0.406452i	\(0.866766\pi\)
\(3\)	0.154854	+	1.72511i	0.0894052	+	0.995995i
							\(5\)	−2.60400	−	0.697740i	−1.16455	−	0.312039i	−0.375766	−	0.926715i	\(-0.622620\pi\)
−0.788780	+	0.614676i	\(0.789287\pi\)
\(7\)	−2.18636	−	0.585835i	−0.826368	−	0.221425i	−0.179240	−	0.983805i	\(-0.557364\pi\)
							−0.647129	+	0.762381i	\(0.724030\pi\)
\(11\)	−1.09462	−	1.09462i	−0.330041	−	0.330041i	0.522561	−	0.852602i	\(-0.324977\pi\)
							−0.852602	+	0.522561i	\(0.824977\pi\)
\(13\)	0.615605	+	3.55261i	0.170738	+	0.985316i
							\(17\)	1.06695	−	1.84800i	0.258772	−	0.448207i	−0.707141	−	0.707073i	\(-0.750015\pi\)
0.965913	+	0.258866i	\(0.0833487\pi\)
\(19\)	−3.84372	+	1.02992i	−0.881809	+	0.236280i	−0.671187	−	0.741288i	\(-0.734215\pi\)
							−0.210622	+	0.977568i	\(0.567549\pi\)
\(23\)	−3.55589	+	6.15899i	−0.741455	+	1.28424i	0.210378	+	0.977620i	\(0.432530\pi\)
							−0.951833	+	0.306617i	\(0.900803\pi\)
\(29\)		−	0.234733i		−	0.0435889i	−0.999762	−	0.0217944i	\(-0.993062\pi\)
							0.999762	−	0.0217944i	\(-0.00693793\pi\)
\(31\)	−1.61398	+	6.02345i	−0.289879	+	1.08184i	0.655321	+	0.755351i	\(0.272534\pi\)
							−0.945200	+	0.326493i	\(0.894133\pi\)
\(37\)	2.21625	+	0.593843i	0.364349	+	0.0976271i	0.436349	−	0.899778i	\(-0.356271\pi\)
							−0.0719995	+	0.997405i	\(0.522938\pi\)
\(41\)	−2.19405	−	8.18830i	−0.342653	−	1.27880i	−0.895331	−	0.445402i	\(-0.853061\pi\)
							0.552678	−	0.833395i	\(-0.313606\pi\)
\(43\)	−0.659047	+	0.380501i	−0.100504	+	0.0580259i	−0.549409	−	0.835553i	\(-0.685147\pi\)
							0.448906	+	0.893579i	\(0.351814\pi\)
\(47\)	−2.39009	+	0.640422i	−0.348630	+	0.0934152i	−0.428885	−	0.903359i	\(-0.641093\pi\)
							0.0802545	+	0.996774i	\(0.474427\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.x.a.11.1	✓	48
3.2	odd	2		351.2.ba.a.89.12		48
9.4	even	3		351.2.bf.a.206.1		48
9.5	odd	6		117.2.bc.a.50.12	yes	48
13.6	odd	12		117.2.bc.a.110.12	yes	48
39.32	even	12		351.2.bf.a.305.1		48
117.32	even	12	inner	117.2.x.a.32.1	yes	48
117.58	odd	12		351.2.ba.a.71.12		48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.x.a.11.1	✓	48	1.1	even	1	trivial
117.2.x.a.32.1	yes	48	117.32	even	12	inner
117.2.bc.a.50.12	yes	48	9.5	odd	6
117.2.bc.a.110.12	yes	48	13.6	odd	12
351.2.ba.a.71.12		48	117.58	odd	12
351.2.ba.a.89.12		48	3.2	odd	2
351.2.bf.a.206.1		48	9.4	even	3
351.2.bf.a.305.1		48	39.32	even	12