Embedded newform 847.2.f.t.323.2

• Label: 847.2.f.t.323.2
• Level: $847$
• Weight: $2$
• Character: 847.323
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.f (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(3\) over \(\Q(\zeta_{5})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	x^12 - x^11 + 7*x^10 - 15*x^9 + 59*x^8 + 118*x^7 + 266*x^6 + 324*x^5 + 1036*x^4 + 376*x^3 + 136*x^2 + 48*x + 16
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			323.2
Root			\(0.112275 + 0.345546i\) of defining polynomial
Character	\(\chi\)	\(=\)	847.323
Dual form			847.2.f.t.729.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10296 - 0.801344i) q^{2} +(0.112275 - 0.345546i) q^{3} +(-0.0436753 - 0.134419i) q^{4} +(-2.54139 + 1.84643i) q^{5} +(-0.400735 + 0.291151i) q^{6} +(0.309017 + 0.951057i) q^{7} +(-0.902127 + 2.77646i) q^{8} +(2.32025 + 1.68576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} + q^{3} - 8 q^{4} - q^{5} - 12 q^{6} - 3 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10296	−	0.801344i	−0.779907	−	0.566636i	0.125044	−	0.992151i	\(-0.460093\pi\)
							−0.904951	+	0.425515i	\(0.860093\pi\)
\(3\)	0.112275	−	0.345546i	0.0648218	−	0.199501i	−0.913400	−	0.407063i	\(-0.866553\pi\)
							0.978222	+	0.207562i	\(0.0665529\pi\)
\(5\)	−2.54139	+	1.84643i	−1.13655	+	0.825749i	−0.986634	−	0.162950i	\(-0.947899\pi\)
							−0.149912	+	0.988699i	\(0.547899\pi\)
\(7\)	0.309017	+	0.951057i	0.116797	+	0.359466i
							\(11\)		0			0
\(13\)	−3.86549	−	2.80844i	−1.07209	−	0.778922i								−0.0958065	−	0.995400i	\(-0.530543\pi\)
							−0.976287	+	0.216478i	\(0.930543\pi\)
\(17\)	−3.86549	+	2.80844i	−0.937519	+	0.681147i	−0.947822	−	0.318800i	\(-0.896720\pi\)
							0.0103032	+	0.999947i	\(0.496720\pi\)
\(19\)	2.16600	−	6.66627i	0.496915	−	1.52935i	−0.317036	−	0.948413i	\(-0.602688\pi\)
							0.813951	−	0.580933i	\(-0.197312\pi\)
\(23\)	5.14134			1.07204			0.536021	−	0.844204i	\(-0.319927\pi\)
							0.536021	+	0.844204i	\(0.319927\pi\)
\(29\)	−2.16600	−	6.66627i	−0.402216	−	1.23789i	−0.923197	−	0.384326i	\(-0.874434\pi\)
							0.520981	−	0.853568i	\(-0.325566\pi\)
\(31\)	2.94213	+	2.13758i	0.528422	+	0.383921i	0.819767	−	0.572697i	\(-0.194103\pi\)
							−0.291345	+	0.956618i	\(0.594103\pi\)
\(37\)	−3.04938	−	9.38502i	−0.501315	−	1.54289i	−0.806879	−	0.590716i	\(-0.798845\pi\)
							0.305565	−	0.952171i	\(-0.401155\pi\)
\(41\)	0.995650	−	3.06430i	0.155494	−	0.478563i	−0.842716	−	0.538358i	\(-0.819045\pi\)
							0.998211	+	0.0597953i	\(0.0190448\pi\)
\(43\)	−4.28267			−0.653101			−0.326551	−	0.945180i	\(-0.605886\pi\)
							−0.326551	+	0.945180i	\(0.605886\pi\)
\(47\)	−0.240418	+	0.739929i	−0.0350685	+	0.107930i	−0.967059	−	0.254554i	\(-0.918071\pi\)
							0.931990	+	0.362484i	\(0.118071\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.f.t.323.2		12
11.2	odd	10		847.2.f.u.148.2		12
11.3	even	5	inner	847.2.f.t.729.2		12
11.4	even	5	inner	847.2.f.t.372.2		12
11.5	even	5		847.2.a.j.1.2	yes	3
11.6	odd	10		847.2.a.i.1.2	✓	3
11.7	odd	10		847.2.f.u.372.2		12
11.8	odd	10		847.2.f.u.729.2		12
11.9	even	5	inner	847.2.f.t.148.2		12
11.10	odd	2		847.2.f.u.323.2		12
33.5	odd	10		7623.2.a.bz.1.2		3
33.17	even	10		7623.2.a.ce.1.2		3
77.6	even	10		5929.2.a.t.1.2		3
77.27	odd	10		5929.2.a.y.1.2		3

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.i.1.2	✓	3	11.6	odd	10
847.2.a.j.1.2	yes	3	11.5	even	5
847.2.f.t.148.2		12	11.9	even	5	inner
847.2.f.t.323.2		12	1.1	even	1	trivial
847.2.f.t.372.2		12	11.4	even	5	inner
847.2.f.t.729.2		12	11.3	even	5	inner
847.2.f.u.148.2		12	11.2	odd	10
847.2.f.u.323.2		12	11.10	odd	2
847.2.f.u.372.2		12	11.7	odd	10
847.2.f.u.729.2		12	11.8	odd	10
5929.2.a.t.1.2		3	77.6	even	10
5929.2.a.y.1.2		3	77.27	odd	10
7623.2.a.bz.1.2		3	33.5	odd	10
7623.2.a.ce.1.2		3	33.17	even	10