Embedded newform 231.2.l.b.32.8

• Label: 231.2.l.b.32.8
• 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			32.8
Character	\(\chi\)	\(=\)	231.32
Dual form			231.2.l.b.65.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.794537 - 1.37618i) q^{2} +(1.04913 - 1.37816i) q^{3} +(-0.262580 + 0.454801i) q^{4} +(-1.14794 + 0.662766i) q^{5} +(-2.73017 - 0.348784i) q^{6} +(-2.48641 + 0.904309i) q^{7} -2.34363 q^{8} +(-0.798666 - 2.89174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.794537	−	1.37618i	−0.561823	−	0.973106i	−0.997337	−	0.0729241i	\(-0.976767\pi\)
							0.435515	−	0.900182i	\(-0.356566\pi\)
\(3\)	1.04913	−	1.37816i	0.605714	−	0.795683i
							\(5\)	−1.14794	+	0.662766i	−0.513376	+	0.296398i	−0.734220	−	0.678911i	\(-0.762452\pi\)
0.220844	+	0.975309i	\(0.429119\pi\)
\(7\)	−2.48641	+	0.904309i	−0.939774	+	0.341797i
							\(11\)	−2.42226	−	2.26554i	−0.730339	−	0.683085i
\(13\)		−	3.14004i		−	0.870891i								−0.900215	−	0.435446i	\(-0.856591\pi\)
							0.900215	−	0.435446i	\(-0.143409\pi\)
\(17\)	1.28658	−	2.22841i	0.312040	−	0.540470i	−0.666764	−	0.745269i	\(-0.732321\pi\)
							0.978804	+	0.204799i	\(0.0656542\pi\)
\(19\)	3.54016	−	2.04391i	0.812168	−	0.468905i	−0.0355403	−	0.999368i	\(-0.511315\pi\)
							0.847708	+	0.530463i	\(0.177982\pi\)
\(23\)	6.23498	−	3.59977i	1.30008	−	0.750604i	0.319665	−	0.947531i	\(-0.396430\pi\)
							0.980418	+	0.196927i	\(0.0630962\pi\)
\(29\)	4.92380			0.914326			0.457163	−	0.889383i	\(-0.348866\pi\)
							0.457163	+	0.889383i	\(0.348866\pi\)
\(31\)	−0.543085	+	0.940652i	−0.0975410	+	0.168946i	−0.910666	−	0.413143i	\(-0.864431\pi\)
							0.813125	+	0.582089i	\(0.197764\pi\)
\(37\)	−0.551859	−	0.955847i	−0.0907250	−	0.157140i	0.817091	−	0.576508i	\(-0.195585\pi\)
							−0.907816	+	0.419368i	\(0.862252\pi\)
\(41\)	6.61451			1.03301			0.516506	−	0.856283i	\(-0.327232\pi\)
							0.516506	+	0.856283i	\(0.327232\pi\)
\(43\)			5.86457i			0.894339i	0.894449	+	0.447169i	\(0.147568\pi\)
							−0.894449	+	0.447169i	\(0.852432\pi\)
\(47\)	−1.18811	+	0.685956i	−0.173304	+	0.100057i	−0.584143	−	0.811651i	\(-0.698569\pi\)
							0.410839	+	0.911708i	\(0.365236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.l.b.32.8	yes	48
3.2	odd	2	inner	231.2.l.b.32.17	yes	48
7.2	even	3	inner	231.2.l.b.65.7	yes	48
11.10	odd	2	inner	231.2.l.b.32.18	yes	48
21.2	odd	6	inner	231.2.l.b.65.18	yes	48
33.32	even	2	inner	231.2.l.b.32.7	✓	48
77.65	odd	6	inner	231.2.l.b.65.17	yes	48
231.65	even	6	inner	231.2.l.b.65.8	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.l.b.32.7	✓	48	33.32	even	2	inner
231.2.l.b.32.8	yes	48	1.1	even	1	trivial
231.2.l.b.32.17	yes	48	3.2	odd	2	inner
231.2.l.b.32.18	yes	48	11.10	odd	2	inner
231.2.l.b.65.7	yes	48	7.2	even	3	inner
231.2.l.b.65.8	yes	48	231.65	even	6	inner
231.2.l.b.65.17	yes	48	77.65	odd	6	inner
231.2.l.b.65.18	yes	48	21.2	odd	6	inner