Embedded newform 287.2.be.a.12.3

• Label: 287.2.be.a.12.3
• 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			12.3
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.75506 - 1.42122i) q^{2} +(-1.39725 + 0.183952i) q^{3} +(0.644551 + 3.03238i) q^{4} +(-1.12109 - 0.0587539i) q^{5} +(2.71371 + 1.66296i) q^{6} +(-1.82427 - 1.91626i) q^{7} +(1.12792 - 2.21366i) q^{8} +(-0.979297 + 0.262402i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.75506	−	1.42122i	−1.24102	−	1.00496i	−0.999402	−	0.0345839i	\(-0.988989\pi\)
							−0.241615	−	0.970372i	\(-0.577677\pi\)
\(3\)	−1.39725	+	0.183952i	−0.806705	+	0.106205i	−0.522588	−	0.852585i	\(-0.675034\pi\)
							−0.284116	+	0.958790i	\(0.591700\pi\)
\(5\)	−1.12109	−	0.0587539i	−0.501367	−	0.0262755i	−0.200023	−	0.979791i	\(-0.564102\pi\)
							−0.301344	+	0.953516i	\(0.597435\pi\)
\(7\)	−1.82427	−	1.91626i	−0.689508	−	0.724278i
							\(11\)	2.16566	+	4.54041i	0.652972	+	1.36898i	0.914815	+	0.403874i	\(0.132337\pi\)
−0.261842	+	0.965111i	\(0.584330\pi\)
\(13\)	−0.638326	−	2.65882i	−0.177040	−	0.737424i	−0.988273	−	0.152701i	\(-0.951203\pi\)
							0.811233	−	0.584724i	\(-0.198797\pi\)
\(17\)	4.48670	+	1.58882i	1.08818	+	0.385346i	0.816680	−	0.577091i	\(-0.195812\pi\)
							0.271505	+	0.962437i	\(0.412479\pi\)
\(19\)	0.910765	+	0.864283i	0.208944	+	0.198280i	0.785650	−	0.618672i	\(-0.212329\pi\)
							−0.576706	+	0.816952i	\(0.695662\pi\)
\(23\)	4.52709	−	0.475816i	0.943963	−	0.0992145i	0.379974	−	0.924997i	\(-0.375933\pi\)
							0.563988	+	0.825783i	\(0.309266\pi\)
\(29\)	−0.977855	+	1.14492i	−0.181583	+	0.212607i	−0.843730	−	0.536767i	\(-0.819645\pi\)
							0.662147	+	0.749374i	\(0.269645\pi\)
\(31\)	4.23266	−	4.70085i	0.760208	−	0.844297i	−0.231496	−	0.972836i	\(-0.574362\pi\)
							0.991704	+	0.128539i	\(0.0410287\pi\)
\(37\)	3.59821	+	3.99622i	0.591542	+	0.656974i	0.962375	−	0.271726i	\(-0.0875944\pi\)
							−0.370833	+	0.928700i	\(0.620928\pi\)
\(41\)	3.33356	+	5.46694i	0.520614	+	0.853792i
							\(43\)	0.460238	−	2.90583i	0.0701856	−	0.443134i	−0.927423	−	0.374015i	\(-0.877981\pi\)
0.997608	−	0.0691197i	\(-0.0220190\pi\)
\(47\)	7.63406	−	4.14496i	1.11354	−	0.604604i	0.187591	−	0.982247i	\(-0.439932\pi\)
							0.925951	+	0.377643i	\(0.123265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.3	✓	832
7.3	odd	6	inner	287.2.be.a.94.24	yes	832
41.24	odd	40	inner	287.2.be.a.229.24	yes	832
287.24	even	120	inner	287.2.be.a.24.3	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.3	✓	832	1.1	even	1	trivial
287.2.be.a.24.3	yes	832	287.24	even	120	inner
287.2.be.a.94.24	yes	832	7.3	odd	6	inner
287.2.be.a.229.24	yes	832	41.24	odd	40	inner