Embedded newform 847.2.p.b.76.14

• Label: 847.2.p.b.76.14
• Level: $847$
• Weight: $2$
• Character: 847.76
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $840$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.p (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(84\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			76.14
Character	\(\chi\)	\(=\)	847.76
Dual form			847.2.p.b.769.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62088 + 1.40450i) q^{2} +1.65558i q^{3} +(0.370000 - 2.57341i) q^{4} +(0.766471 - 1.19265i) q^{5} +(-2.32526 - 2.68350i) q^{6} +(-2.61253 + 0.417949i) q^{7} +(0.695569 + 1.08233i) q^{8} +0.259056 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 22 q^{2} + 58 q^{4} - 11 q^{7} - 22 q^{8} - 924 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62088	+	1.40450i	−1.14614	+	0.993132i	−0.146143	+	0.989264i	\(0.546686\pi\)
							−0.999993	+	0.00386813i	\(0.998769\pi\)
\(3\)			1.65558i			0.955849i	0.878401	+	0.477925i	\(0.158611\pi\)
							−0.878401	+	0.477925i	\(0.841389\pi\)
\(5\)	0.766471	−	1.19265i	0.342776	−	0.533370i	−0.626476	−	0.779441i	\(-0.715503\pi\)
							0.969252	+	0.246071i	\(0.0791396\pi\)
\(7\)	−2.61253	+	0.417949i	−0.987444	+	0.157970i
							\(11\)	0.853720	−	3.20487i	0.257406	−	0.966303i
\(13\)	0.205417	−	1.42871i	0.0569724	−	0.396252i								−0.941304	−	0.337561i	\(-0.890398\pi\)
							0.998276	−	0.0586911i	\(-0.0186927\pi\)
\(17\)	1.85557	+	4.06313i	0.450042	+	0.985454i	0.989645	+	0.143535i	\(0.0458470\pi\)
							−0.539603	+	0.841919i	\(0.681426\pi\)
\(19\)	0.336983	−	0.737890i	0.0773092	−	0.169284i	−0.867031	−	0.498254i	\(-0.833975\pi\)
							0.944340	+	0.328971i	\(0.106702\pi\)
\(23\)	−4.53593	−	1.33187i	−0.945808	−	0.277714i	−0.227767	−	0.973716i	\(-0.573143\pi\)
							−0.718040	+	0.696001i	\(0.754961\pi\)
\(29\)	4.40447	+	2.01145i	0.817889	+	0.373517i	0.780012	−	0.625765i	\(-0.215213\pi\)
							0.0378772	+	0.999282i	\(0.487940\pi\)
\(31\)	3.93840	−	0.566256i	0.707357	−	0.101703i	0.220759	−	0.975328i	\(-0.429147\pi\)
							0.486598	+	0.873626i	\(0.338238\pi\)
\(37\)	0.0688447	+	0.478826i	0.0113180	+	0.0787185i	0.994698	−	0.102843i	\(-0.0327939\pi\)
							−0.983380	+	0.181562i	\(0.941885\pi\)
\(41\)	0.344609	+	0.397700i	0.0538189	+	0.0621103i	0.782021	−	0.623252i	\(-0.214189\pi\)
							−0.728202	+	0.685363i	\(0.759644\pi\)
\(43\)	1.91698	+	2.98288i	0.292336	+	0.454884i	0.956093	−	0.293065i	\(-0.0946751\pi\)
							−0.663756	+	0.747949i	\(0.731039\pi\)
\(47\)	8.69543	+	7.53463i	1.26836	+	1.09904i	0.990369	+	0.138455i	\(0.0442135\pi\)
							0.277990	+	0.960584i	\(0.410332\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.p.b.76.14	yes	840
7.6	odd	2	inner	847.2.p.b.76.13	✓	840
121.43	odd	22	inner	847.2.p.b.769.14	yes	840
847.769	even	22	inner	847.2.p.b.769.13	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.p.b.76.13	✓	840	7.6	odd	2	inner
847.2.p.b.76.14	yes	840	1.1	even	1	trivial
847.2.p.b.769.13	yes	840	847.769	even	22	inner
847.2.p.b.769.14	yes	840	121.43	odd	22	inner