Embedded newform 231.2.l.b.32.6

• Label: 231.2.l.b.32.6
• Level: $231$
• Weight: $2$
• Character: 231.32
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			32.6
Character	\(\chi\)	\(=\)	231.32
Dual form			231.2.l.b.65.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.939478 - 1.62722i) q^{2} +(1.16688 + 1.27999i) q^{3} +(-0.765237 + 1.32543i) q^{4} +(-2.63713 + 1.52255i) q^{5} +(0.986571 - 3.10131i) q^{6} +(0.738610 + 2.54056i) q^{7} -0.882219 q^{8} +(-0.276760 + 2.98721i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\)	\(155\)	\(199\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.939478	−	1.62722i	−0.664311	−	1.15062i	−0.979472	−	0.201582i	\(-0.935392\pi\)
							0.315161	−	0.949038i	\(-0.397942\pi\)
\(3\)	1.16688	+	1.27999i	0.673701	+	0.739004i
							\(5\)	−2.63713	+	1.52255i	−1.17936	+	0.680904i	−0.955867	−	0.293801i	\(-0.905080\pi\)
−0.223494	+	0.974705i	\(0.571746\pi\)
\(7\)	0.738610	+	2.54056i	0.279168	+	0.960242i
							\(11\)	3.23606	−	0.726556i	0.975710	−	0.219065i
\(13\)			1.24436i			0.345123i								0.984999	+	0.172561i	\(0.0552043\pi\)
							−0.984999	+	0.172561i	\(0.944796\pi\)
\(17\)	−2.83407	+	4.90876i	−0.687363	+	1.19055i	0.285325	+	0.958431i	\(0.407899\pi\)
							−0.972688	+	0.232117i	\(0.925435\pi\)
\(19\)	−0.532275	+	0.307309i	−0.122112	+	0.0705015i	−0.559812	−	0.828620i	\(-0.689127\pi\)
							0.437700	+	0.899121i	\(0.355793\pi\)
\(23\)	4.41744	−	2.55041i	0.921101	−	0.531798i	0.0371147	−	0.999311i	\(-0.488183\pi\)
							0.883986	+	0.467513i	\(0.154850\pi\)
\(29\)	−1.47768			−0.274398			−0.137199	−	0.990544i	\(-0.543810\pi\)
							−0.137199	+	0.990544i	\(0.543810\pi\)
\(31\)	3.18007	−	5.50805i	0.571158	−	0.989275i	−0.425289	−	0.905057i	\(-0.639828\pi\)
							0.996447	−	0.0842172i	\(-0.0268390\pi\)
\(37\)	−3.15832	−	5.47036i	−0.519224	−	0.899322i	−0.999750	−	0.0223421i	\(-0.992888\pi\)
							0.480526	−	0.876980i	\(-0.340446\pi\)
\(41\)	10.3640			1.61859			0.809296	−	0.587401i	\(-0.199849\pi\)
							0.809296	+	0.587401i	\(0.199849\pi\)
\(43\)		−	2.31140i		−	0.352486i	−0.984347	−	0.176243i	\(-0.943606\pi\)
							0.984347	−	0.176243i	\(-0.0563944\pi\)
\(47\)	−9.35776	+	5.40270i	−1.36497	+	0.788065i	−0.990280	−	0.139086i	\(-0.955584\pi\)
							−0.374689	+	0.927151i	\(0.622250\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.l.b.32.6	yes	48
3.2	odd	2	inner	231.2.l.b.32.19	yes	48
7.2	even	3	inner	231.2.l.b.65.5	yes	48
11.10	odd	2	inner	231.2.l.b.32.20	yes	48
21.2	odd	6	inner	231.2.l.b.65.20	yes	48
33.32	even	2	inner	231.2.l.b.32.5	✓	48
77.65	odd	6	inner	231.2.l.b.65.19	yes	48
231.65	even	6	inner	231.2.l.b.65.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.l.b.32.5	✓	48	33.32	even	2	inner
231.2.l.b.32.6	yes	48	1.1	even	1	trivial
231.2.l.b.32.19	yes	48	3.2	odd	2	inner
231.2.l.b.32.20	yes	48	11.10	odd	2	inner
231.2.l.b.65.5	yes	48	7.2	even	3	inner
231.2.l.b.65.6	yes	48	231.65	even	6	inner
231.2.l.b.65.19	yes	48	77.65	odd	6	inner
231.2.l.b.65.20	yes	48	21.2	odd	6	inner