Embedded newform 231.2.l.b.32.9

• Label: 231.2.l.b.32.9
• 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			32.9
Character	\(\chi\)	\(=\)	231.32
Dual form			231.2.l.b.65.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.468581 - 0.811606i) q^{2} +(-1.03189 + 1.39112i) q^{3} +(0.560864 - 0.971445i) q^{4} +(2.73832 - 1.58097i) q^{5} +(1.61256 + 0.185636i) q^{6} +(-0.0402023 + 2.64545i) q^{7} -2.92556 q^{8} +(-0.870410 - 2.87096i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{3} - 36 q^{4} - 10 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.468581	−	0.811606i	−0.331337	−	0.573892i	0.651438	−	0.758702i	\(-0.274166\pi\)
							−0.982774	+	0.184810i	\(0.940833\pi\)
\(3\)	−1.03189	+	1.39112i	−0.595761	+	0.803162i
							\(5\)	2.73832	−	1.58097i	1.22461	−	0.707031i	0.258715	−	0.965954i	\(-0.416701\pi\)
0.965898	+	0.258923i	\(0.0833676\pi\)
\(7\)	−0.0402023	+	2.64545i	−0.0151950	+	0.999885i
							\(11\)	2.99764	−	1.41920i	0.903823	−	0.427906i
\(13\)			1.78405i			0.494807i								0.968913	+	0.247403i	\(0.0795773\pi\)
							−0.968913	+	0.247403i	\(0.920423\pi\)
\(17\)	2.35195	−	4.07369i	0.570431	−	0.988015i	−0.426091	−	0.904680i	\(-0.640110\pi\)
							0.996522	−	0.0833349i	\(-0.0265571\pi\)
\(19\)	6.24385	−	3.60489i	1.43244	−	0.827018i	0.435131	−	0.900367i	\(-0.356702\pi\)
							0.997306	+	0.0733490i	\(0.0233687\pi\)
\(23\)	−0.462754	+	0.267171i	−0.0964908	+	0.0557090i	−0.547469	−	0.836826i	\(-0.684409\pi\)
							0.450978	+	0.892535i	\(0.351075\pi\)
\(29\)	−6.46212			−1.19999			−0.599993	−	0.800006i	\(-0.704830\pi\)
							−0.599993	+	0.800006i	\(0.704830\pi\)
\(31\)	−1.71277	+	2.96661i	−0.307623	+	0.532818i	−0.977842	−	0.209345i	\(-0.932867\pi\)
							0.670219	+	0.742163i	\(0.266200\pi\)
\(37\)	2.08135	+	3.60500i	0.342172	+	0.592658i	0.984836	−	0.173490i	\(-0.0555043\pi\)
							−0.642664	+	0.766148i	\(0.722171\pi\)
\(41\)	−0.272850			−0.0426121			−0.0213060	−	0.999773i	\(-0.506782\pi\)
							−0.0213060	+	0.999773i	\(0.506782\pi\)
\(43\)			5.32723i			0.812395i	0.913785	+	0.406198i	\(0.133146\pi\)
							−0.913785	+	0.406198i	\(0.866854\pi\)
\(47\)	−7.52310	+	4.34347i	−1.09736	+	0.633560i	−0.935526	−	0.353259i	\(-0.885074\pi\)
							−0.161832	+	0.986818i	\(0.551740\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.l.b.32.9	✓	48
3.2	odd	2	inner	231.2.l.b.32.16	yes	48
7.2	even	3	inner	231.2.l.b.65.10	yes	48
11.10	odd	2	inner	231.2.l.b.32.15	yes	48
21.2	odd	6	inner	231.2.l.b.65.15	yes	48
33.32	even	2	inner	231.2.l.b.32.10	yes	48
77.65	odd	6	inner	231.2.l.b.65.16	yes	48
231.65	even	6	inner	231.2.l.b.65.9	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.l.b.32.9	✓	48	1.1	even	1	trivial
231.2.l.b.32.10	yes	48	33.32	even	2	inner
231.2.l.b.32.15	yes	48	11.10	odd	2	inner
231.2.l.b.32.16	yes	48	3.2	odd	2	inner
231.2.l.b.65.9	yes	48	231.65	even	6	inner
231.2.l.b.65.10	yes	48	7.2	even	3	inner
231.2.l.b.65.15	yes	48	21.2	odd	6	inner
231.2.l.b.65.16	yes	48	77.65	odd	6	inner