Embedded newform 231.2.l.b.32.12

• Label: 231.2.l.b.32.12
• 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			32.12
Character	\(\chi\)	\(=\)	231.32
Dual form			231.2.l.b.65.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.463987 - 0.803650i) q^{2} +(1.07701 + 1.35648i) q^{3} +(0.569432 - 0.986284i) q^{4} +(1.53501 - 0.886239i) q^{5} +(0.590414 - 1.49493i) q^{6} +(2.26798 - 1.36244i) q^{7} -2.91279 q^{8} +(-0.680079 + 2.92190i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.463987	−	0.803650i	−0.328089	−	0.568266i	0.654044	−	0.756457i	\(-0.273071\pi\)
							−0.982133	+	0.188190i	\(0.939738\pi\)
\(3\)	1.07701	+	1.35648i	0.621815	+	0.783164i
							\(5\)	1.53501	−	0.886239i	0.686477	−	0.396338i	−0.115814	−	0.993271i	\(-0.536948\pi\)
0.802291	+	0.596933i	\(0.203614\pi\)
\(7\)	2.26798	−	1.36244i	0.857217	−	0.514956i
							\(11\)	−3.22069	+	0.791945i	−0.971074	+	0.238780i
\(13\)		−	0.151989i		−	0.0421542i								−0.999778	−	0.0210771i	\(-0.993290\pi\)
							0.999778	−	0.0210771i	\(-0.00670955\pi\)
\(17\)	−0.598686	+	1.03695i	−0.145203	+	0.251498i	−0.929449	−	0.368952i	\(-0.879717\pi\)
							0.784246	+	0.620450i	\(0.213050\pi\)
\(19\)	4.13286	−	2.38611i	0.948144	−	0.547411i	0.0556399	−	0.998451i	\(-0.482280\pi\)
							0.892504	+	0.451040i	\(0.148947\pi\)
\(23\)	1.52546	−	0.880725i	0.318080	−	0.183644i	−0.332456	−	0.943119i	\(-0.607877\pi\)
							0.650537	+	0.759475i	\(0.274544\pi\)
\(29\)	5.83464			1.08346			0.541732	−	0.840551i	\(-0.317769\pi\)
							0.541732	+	0.840551i	\(0.317769\pi\)
\(31\)	−2.85942	+	4.95267i	−0.513568	+	0.889525i	0.486309	+	0.873787i	\(0.338343\pi\)
							−0.999876	+	0.0157380i	\(0.994990\pi\)
\(37\)	−4.12359	−	7.14227i	−0.677914	−	1.17418i	−0.975608	−	0.219520i	\(-0.929551\pi\)
							0.297694	−	0.954661i	\(-0.403783\pi\)
\(41\)	−10.5905			−1.65395			−0.826975	−	0.562238i	\(-0.809940\pi\)
							−0.826975	+	0.562238i	\(0.809940\pi\)
\(43\)			10.5255i			1.60513i	0.596564	+	0.802566i	\(0.296532\pi\)
							−0.596564	+	0.802566i	\(0.703468\pi\)
\(47\)	1.28151	−	0.739881i	0.186928	−	0.107923i	−0.403616	−	0.914929i	\(-0.632247\pi\)
							0.590543	+	0.807006i	\(0.298913\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.l.b.32.12	yes	48
3.2	odd	2	inner	231.2.l.b.32.13	yes	48
7.2	even	3	inner	231.2.l.b.65.11	yes	48
11.10	odd	2	inner	231.2.l.b.32.14	yes	48
21.2	odd	6	inner	231.2.l.b.65.14	yes	48
33.32	even	2	inner	231.2.l.b.32.11	✓	48
77.65	odd	6	inner	231.2.l.b.65.13	yes	48
231.65	even	6	inner	231.2.l.b.65.12	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.l.b.32.11	✓	48	33.32	even	2	inner
231.2.l.b.32.12	yes	48	1.1	even	1	trivial
231.2.l.b.32.13	yes	48	3.2	odd	2	inner
231.2.l.b.32.14	yes	48	11.10	odd	2	inner
231.2.l.b.65.11	yes	48	7.2	even	3	inner
231.2.l.b.65.12	yes	48	231.65	even	6	inner
231.2.l.b.65.13	yes	48	77.65	odd	6	inner
231.2.l.b.65.14	yes	48	21.2	odd	6	inner