Embedded newform 847.2.be.a.17.11

• Label: 847.2.be.a.17.11
• Level: $847$
• Weight: $2$
• Character: 847.17
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $6880$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.be (of order \(330\), degree \(80\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(6880\)
Relative dimension:	\(86\) over \(\Q(\zeta_{330})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{330}]$

Embedding invariants

Embedding label			17.11
Character	\(\chi\)	\(=\)	847.17
Dual form			847.2.be.a.299.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.73569 + 1.44706i) q^{2} +(-0.981739 - 2.20502i) q^{3} +(0.558861 - 3.05592i) q^{4} +(-2.62084 + 1.94254i) q^{5} +(4.89479 + 2.40660i) q^{6} +(2.62119 - 0.359701i) q^{7} +(1.22966 + 2.17744i) q^{8} +(-1.89092 + 2.10008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6880 q - 39 q^{2} - 90 q^{3} + 45 q^{4} - 117 q^{5} - 83 q^{7} - 176 q^{8} - 844 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{49}{110}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.73569	+	1.44706i	−1.22732	+	1.02322i	−0.228554	+	0.973531i	\(0.573400\pi\)
							−0.998765	+	0.0496926i	\(0.984176\pi\)
\(3\)	−0.981739	−	2.20502i	−0.566807	−	1.27307i	−0.938681	−	0.344787i	\(-0.887951\pi\)
							0.371874	−	0.928283i	\(-0.378715\pi\)
\(5\)	−2.62084	+	1.94254i	−1.17207	+	0.868730i	−0.993346	−	0.115169i	\(-0.963259\pi\)
							−0.178729	+	0.983898i	\(0.557199\pi\)
\(7\)	2.62119	−	0.359701i	0.990715	−	0.135954i
							\(11\)	−3.14311	−	1.05872i	−0.947682	−	0.319216i
\(13\)	0.0474887	+	1.66232i	0.0131710	+	0.461045i								0.979233	+	0.202736i	\(0.0649834\pi\)
							−0.966062	+	0.258309i	\(0.916835\pi\)
\(17\)	4.17370	−	0.0794767i	1.01227	−	0.0192759i	0.489717	−	0.871882i	\(-0.337100\pi\)
							0.522555	+	0.852606i	\(0.324979\pi\)
\(19\)	0.826829	−	0.576999i	0.189688	−	0.132373i	−0.473592	−	0.880744i	\(-0.657043\pi\)
							0.663280	+	0.748372i	\(0.269164\pi\)
\(23\)	−0.276165	+	5.79742i	−0.0575844	+	1.20885i	0.766449	+	0.642305i	\(0.222022\pi\)
							−0.824034	+	0.566541i	\(0.808281\pi\)
\(29\)	0.164723	−	0.708363i	0.0305882	−	0.131540i	−0.956495	−	0.291750i	\(-0.905763\pi\)
							0.987083	+	0.160210i	\(0.0512171\pi\)
\(31\)	0.210876	+	2.76351i	0.0378745	+	0.496340i	0.984661	+	0.174478i	\(0.0558239\pi\)
							−0.946787	+	0.321862i	\(0.895691\pi\)
\(37\)	4.16071	+	0.557845i	0.684016	+	0.0917092i	0.466732	−	0.884399i	\(-0.345431\pi\)
							0.217285	+	0.976108i	\(0.430280\pi\)
\(41\)	1.75377	−	1.80459i	0.273893	−	0.281829i	−0.566204	−	0.824265i	\(-0.691589\pi\)
							0.840097	+	0.542436i	\(0.182498\pi\)
\(43\)	−10.0238	−	4.57771i	−1.52861	−	0.698095i	−0.539062	−	0.842266i	\(-0.681221\pi\)
							−0.989553	+	0.144171i	\(0.953948\pi\)
\(47\)	−0.399806	−	10.4941i	−0.0583177	−	1.53072i	−0.671637	−	0.740880i	\(-0.734409\pi\)
							0.613319	−	0.789835i	\(-0.289834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.be.a.17.11	✓	6880
7.5	odd	6	inner	847.2.be.a.138.76	yes	6880
121.57	odd	110	inner	847.2.be.a.178.76	yes	6880
847.299	even	330	inner	847.2.be.a.299.11	yes	6880

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.be.a.17.11	✓	6880	1.1	even	1	trivial
847.2.be.a.138.76	yes	6880	7.5	odd	6	inner
847.2.be.a.178.76	yes	6880	121.57	odd	110	inner
847.2.be.a.299.11	yes	6880	847.299	even	330	inner