Embedded newform 287.2.be.a.12.8

• Label: 287.2.be.a.12.8
• Level: $287$
• Weight: $2$
• Character: 287.12
• Analytic conductor: $2.292$
• Analytic rank: $0$
• Dimension: $832$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	287.be (of order \(120\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.29170653801\)
Analytic rank:	\(0\)
Dimension:	\(832\)
Relative dimension:	\(26\) over \(\Q(\zeta_{120})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{120}]$

Embedding invariants

Embedding label			12.8
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981014 - 0.794410i) q^{2} +(-2.00494 + 0.263955i) q^{3} +(-0.0845211 - 0.397641i) q^{4} +(-3.10557 - 0.162756i) q^{5} +(2.17656 + 1.33380i) q^{6} +(1.34026 - 2.28116i) q^{7} +(-1.37914 + 2.70672i) q^{8} +(1.05233 - 0.281971i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{27}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.981014	−	0.794410i	−0.693682	−	0.561732i	0.216327	−	0.976321i	\(-0.430592\pi\)
							−0.910009	+	0.414588i	\(0.863926\pi\)
\(3\)	−2.00494	+	0.263955i	−1.15755	+	0.152395i	−0.684753	−	0.728776i	\(-0.740090\pi\)
							−0.472799	+	0.881170i	\(0.656756\pi\)
\(5\)	−3.10557	−	0.162756i	−1.38885	−	0.0727868i	−0.656816	−	0.754050i	\(-0.728097\pi\)
							−0.732038	+	0.681264i	\(0.761431\pi\)
\(7\)	1.34026	−	2.28116i	0.506571	−	0.862198i
							\(11\)	−1.93643	−	4.05980i	−0.583854	−	1.22408i	−0.955305	−	0.295623i	\(-0.904473\pi\)
0.371450	−	0.928453i	\(-0.378861\pi\)
\(13\)	1.43291	+	5.96848i	0.397417	+	1.65536i	0.709051	+	0.705158i	\(0.249124\pi\)
							−0.311634	+	0.950202i	\(0.600876\pi\)
\(17\)	0.445641	+	0.157810i	0.108084	+	0.0382745i	0.387300	−	0.921954i	\(-0.373408\pi\)
							−0.279216	+	0.960228i	\(0.590075\pi\)
\(19\)	3.92074	+	3.72064i	0.899480	+	0.853574i	0.989818	−	0.142337i	\(-0.0454616\pi\)
							−0.0903388	+	0.995911i	\(0.528795\pi\)
\(23\)	2.06680	−	0.217229i	0.430957	−	0.0452954i	0.113433	−	0.993546i	\(-0.463815\pi\)
							0.317524	+	0.948250i	\(0.397149\pi\)
\(29\)	−2.36271	+	2.76637i	−0.438743	+	0.513702i	−0.935084	−	0.354425i	\(-0.884677\pi\)
							0.496341	+	0.868128i	\(0.334677\pi\)
\(31\)	−4.57998	+	5.08658i	−0.822588	+	0.913577i	−0.997475	−	0.0710119i	\(-0.977377\pi\)
							0.174887	+	0.984588i	\(0.444044\pi\)
\(37\)	−2.73274	−	3.03502i	−0.449260	−	0.498954i	0.475388	−	0.879776i	\(-0.342308\pi\)
							−0.924648	+	0.380822i	\(0.875641\pi\)
\(41\)	0.553913	−	6.37912i	0.0865067	−	0.996251i
							\(43\)	−1.32402	+	8.35951i	−0.201911	+	1.27481i	0.653526	+	0.756904i	\(0.273289\pi\)
−0.855436	+	0.517908i	\(0.826711\pi\)
\(47\)	2.94053	−	1.59658i	0.428920	−	0.232884i	−0.248273	−	0.968690i	\(-0.579863\pi\)
							0.677193	+	0.735806i	\(0.263196\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.8	✓	832
7.3	odd	6	inner	287.2.be.a.94.19	yes	832
41.24	odd	40	inner	287.2.be.a.229.19	yes	832
287.24	even	120	inner	287.2.be.a.24.8	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.8	✓	832	1.1	even	1	trivial
287.2.be.a.24.8	yes	832	287.24	even	120	inner
287.2.be.a.94.19	yes	832	7.3	odd	6	inner
287.2.be.a.229.19	yes	832	41.24	odd	40	inner