Embedded newform 231.2.l.b.32.2

• Label: 231.2.l.b.32.2
• 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			32.2
Character	\(\chi\)	\(=\)	231.32
Dual form			231.2.l.b.65.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.36539 - 2.36492i) q^{2} +(1.68995 - 0.379546i) q^{3} +(-2.72858 + 4.72604i) q^{4} +(1.96785 - 1.13614i) q^{5} +(-3.20504 - 3.47838i) q^{6} +(1.90462 + 1.83641i) q^{7} +9.44073 q^{8} +(2.71189 - 1.28283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.36539	−	2.36492i	−0.965476	−	1.67225i	−0.708330	−	0.705881i	\(-0.750551\pi\)
							−0.257146	−	0.966372i	\(-0.582782\pi\)
\(3\)	1.68995	−	0.379546i	0.975695	−	0.219131i
							\(5\)	1.96785	−	1.13614i	0.880047	−	0.508095i	0.00937323	−	0.999956i	\(-0.497016\pi\)
0.870674	+	0.491861i	\(0.163683\pi\)
\(7\)	1.90462	+	1.83641i	0.719879	+	0.694099i
							\(11\)	−2.00496	−	2.64200i	−0.604518	−	0.796592i
\(13\)		−	0.322402i		−	0.0894182i								−0.999000	−	0.0447091i	\(-0.985764\pi\)
							0.999000	−	0.0447091i	\(-0.0142361\pi\)
\(17\)	−1.73168	+	2.99936i	−0.419995	+	0.727452i	−0.995938	−	0.0900373i	\(-0.971301\pi\)
							0.575944	+	0.817489i	\(0.304635\pi\)
\(19\)	3.16643	−	1.82814i	0.726428	−	0.419403i	−0.0906861	−	0.995880i	\(-0.528906\pi\)
							0.817114	+	0.576476i	\(0.195573\pi\)
\(23\)	−2.67391	+	1.54378i	−0.557550	+	0.321901i	−0.752161	−	0.658979i	\(-0.770989\pi\)
							0.194612	+	0.980880i	\(0.437655\pi\)
\(29\)	−6.42308			−1.19274			−0.596368	−	0.802711i	\(-0.703390\pi\)
							−0.596368	+	0.802711i	\(0.703390\pi\)
\(31\)	−2.42106	+	4.19340i	−0.434835	+	0.753156i	−0.997282	−	0.0736770i	\(-0.976527\pi\)
							0.562447	+	0.826833i	\(0.309860\pi\)
\(37\)	−0.690076	−	1.19525i	−0.113448	−	0.196497i	0.803710	−	0.595021i	\(-0.202856\pi\)
							−0.917158	+	0.398523i	\(0.869523\pi\)
\(41\)	−3.42022			−0.534149			−0.267074	−	0.963676i	\(-0.586057\pi\)
							−0.267074	+	0.963676i	\(0.586057\pi\)
\(43\)		−	5.82380i		−	0.888122i	−0.895997	−	0.444061i	\(-0.853537\pi\)
							0.895997	−	0.444061i	\(-0.146463\pi\)
\(47\)	2.37383	−	1.37053i	0.346259	−	0.199913i	−0.316777	−	0.948500i	\(-0.602601\pi\)
							0.663036	+	0.748587i	\(0.269267\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.l.b.32.2	yes	48
3.2	odd	2	inner	231.2.l.b.32.23	yes	48
7.2	even	3	inner	231.2.l.b.65.1	yes	48
11.10	odd	2	inner	231.2.l.b.32.24	yes	48
21.2	odd	6	inner	231.2.l.b.65.24	yes	48
33.32	even	2	inner	231.2.l.b.32.1	✓	48
77.65	odd	6	inner	231.2.l.b.65.23	yes	48
231.65	even	6	inner	231.2.l.b.65.2	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.l.b.32.1	✓	48	33.32	even	2	inner
231.2.l.b.32.2	yes	48	1.1	even	1	trivial
231.2.l.b.32.23	yes	48	3.2	odd	2	inner
231.2.l.b.32.24	yes	48	11.10	odd	2	inner
231.2.l.b.65.1	yes	48	7.2	even	3	inner
231.2.l.b.65.2	yes	48	231.65	even	6	inner
231.2.l.b.65.23	yes	48	77.65	odd	6	inner
231.2.l.b.65.24	yes	48	21.2	odd	6	inner