Embedded newform 231.2.u.a.20.14

• Label: 231.2.u.a.20.14
• Level: $231$
• Weight: $2$
• Character: 231.20
• Analytic conductor: $1.845$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	231.u (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.84454428669\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			20.14
Character	\(\chi\)	\(=\)	231.20
Dual form			231.2.u.a.104.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.126891 - 0.0412293i) q^{2} +(0.347648 + 1.69680i) q^{3} +(-1.60363 - 1.16511i) q^{4} +(0.887927 + 2.73276i) q^{5} +(0.0258447 - 0.229642i) q^{6} +(2.60765 - 0.447387i) q^{7} +(0.312296 + 0.429838i) q^{8} +(-2.75828 + 1.17978i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q + 12 q^{4} - 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(155\)	\(199\)	\(211\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.126891	−	0.0412293i	−0.0897254	−	0.0291536i	0.263810	−	0.964575i	\(-0.415021\pi\)
							−0.353536	+	0.935421i	\(0.615021\pi\)
\(3\)	0.347648	+	1.69680i	0.200715	+	0.979650i
							\(5\)	0.887927	+	2.73276i	0.397093	+	1.22213i	0.927319	+	0.374271i	\(0.122107\pi\)
−0.530226	+	0.847856i	\(0.677893\pi\)
\(7\)	2.60765	−	0.447387i	0.985599	−	0.169097i
							\(11\)	−2.78405	+	1.80251i	−0.839423	+	0.543479i
\(13\)	−1.85148	−	0.601582i	−0.513508	−	0.166849i								0.0407892	−	0.999168i	\(-0.487013\pi\)
							−0.554297	+	0.832319i	\(0.687013\pi\)
\(17\)	1.87135	+	5.75941i	0.453868	+	1.39686i	0.872459	+	0.488688i	\(0.162524\pi\)
							−0.418590	+	0.908175i	\(0.637476\pi\)
\(19\)	2.09302	+	2.88080i	0.480173	+	0.660901i	0.978538	−	0.206066i	\(-0.0660661\pi\)
							−0.498365	+	0.866967i	\(0.666066\pi\)
\(23\)		−	5.52810i		−	1.15269i	−0.817207	−	0.576344i	\(-0.804479\pi\)
							0.817207	−	0.576344i	\(-0.195521\pi\)
\(29\)	4.07967	−	5.61519i	0.757576	−	1.04271i	−0.239836	−	0.970814i	\(-0.577094\pi\)
							0.997412	−	0.0719007i	\(-0.0229065\pi\)
\(31\)	3.17461	+	1.03149i	0.570176	+	0.185261i	0.579895	−	0.814691i	\(-0.303094\pi\)
							−0.00971880	+	0.999953i	\(0.503094\pi\)
\(37\)	−4.68486	−	3.40375i	−0.770187	−	0.559573i	0.131831	−	0.991272i	\(-0.457914\pi\)
							−0.902018	+	0.431699i	\(0.857914\pi\)
\(41\)	5.14995	−	3.74166i	0.804287	−	0.584349i	−0.107882	−	0.994164i	\(-0.534407\pi\)
							0.912169	+	0.409815i	\(0.134407\pi\)
\(43\)	2.30910			0.352135			0.176067	−	0.984378i	\(-0.443662\pi\)
							0.176067	+	0.984378i	\(0.443662\pi\)
\(47\)	2.95837	−	2.14938i	0.431522	−	0.313519i	−0.350735	−	0.936475i	\(-0.614068\pi\)
							0.782257	+	0.622955i	\(0.214068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.2.u.a.20.14	yes	112
3.2	odd	2	inner	231.2.u.a.20.16	yes	112
7.6	odd	2	inner	231.2.u.a.20.13	✓	112
11.5	even	5	inner	231.2.u.a.104.15	yes	112
21.20	even	2	inner	231.2.u.a.20.15	yes	112
33.5	odd	10	inner	231.2.u.a.104.13	yes	112
77.27	odd	10	inner	231.2.u.a.104.16	yes	112
231.104	even	10	inner	231.2.u.a.104.14	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.u.a.20.13	✓	112	7.6	odd	2	inner
231.2.u.a.20.14	yes	112	1.1	even	1	trivial
231.2.u.a.20.15	yes	112	21.20	even	2	inner
231.2.u.a.20.16	yes	112	3.2	odd	2	inner
231.2.u.a.104.13	yes	112	33.5	odd	10	inner
231.2.u.a.104.14	yes	112	231.104	even	10	inner
231.2.u.a.104.15	yes	112	11.5	even	5	inner
231.2.u.a.104.16	yes	112	77.27	odd	10	inner