Embedded newform 287.2.be.a.12.13

• Label: 287.2.be.a.12.13
• 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			12.13
Character	\(\chi\)	\(=\)	287.12
Dual form			287.2.be.a.24.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.213546 - 0.172926i) q^{2} +(-1.16847 + 0.153832i) q^{3} +(-0.400125 - 1.88244i) q^{4} +(-0.113709 - 0.00595924i) q^{5} +(0.276122 + 0.169208i) q^{6} +(-2.51661 + 0.816511i) q^{7} +(-0.489574 + 0.960843i) q^{8} +(-1.55613 + 0.416963i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 832 q - 16 q^{2} - 48 q^{3} - 20 q^{4} - 48 q^{5} - 32 q^{7} - 48 q^{8} - 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.213546	−	0.172926i	−0.150999	−	0.122277i	0.550844	−	0.834608i	\(-0.314306\pi\)
							−0.701843	+	0.712331i	\(0.747639\pi\)
\(3\)	−1.16847	+	0.153832i	−0.674615	+	0.0888147i	−0.460047	−	0.887894i	\(-0.652168\pi\)
							−0.214568	+	0.976709i	\(0.568834\pi\)
\(5\)	−0.113709	−	0.00595924i	−0.0508522	−	0.00266505i	0.0268929	−	0.999638i	\(-0.491439\pi\)
							−0.0777451	+	0.996973i	\(0.524772\pi\)
\(7\)	−2.51661	+	0.816511i	−0.951188	+	0.308612i
							\(11\)	1.48969	+	3.12321i	0.449160	+	0.941683i	0.994624	+	0.103549i	\(0.0330199\pi\)
−0.545464	+	0.838134i	\(0.683647\pi\)
\(13\)	0.528221	+	2.20020i	0.146502	+	0.610225i	0.996511	+	0.0834572i	\(0.0265962\pi\)
							−0.850009	+	0.526768i	\(0.823404\pi\)
\(17\)	0.415306	+	0.147068i	0.100727	+	0.0356692i	0.383693	−	0.923461i	\(-0.374652\pi\)
							−0.282967	+	0.959130i	\(0.591319\pi\)
\(19\)	0.876311	+	0.831588i	0.201040	+	0.190779i	0.782233	−	0.622986i	\(-0.214081\pi\)
							−0.581193	+	0.813765i	\(0.697414\pi\)
\(23\)	−6.76368	+	0.710891i	−1.41032	+	0.148231i	−0.778872	−	0.627183i	\(-0.784208\pi\)
							−0.631453	+	0.775414i	\(0.717541\pi\)
\(29\)	−1.40516	+	1.64524i	−0.260933	+	0.305513i	−0.875371	−	0.483452i	\(-0.839383\pi\)
							0.614438	+	0.788965i	\(0.289383\pi\)
\(31\)	−3.44020	+	3.82073i	−0.617879	+	0.686224i	−0.968135	−	0.250430i	\(-0.919428\pi\)
							0.350256	+	0.936654i	\(0.386095\pi\)
\(37\)	−6.96810	−	7.73886i	−1.14555	−	1.27226i	−0.956965	−	0.290204i	\(-0.906277\pi\)
							−0.188584	−	0.982057i	\(-0.560390\pi\)
\(41\)	5.67148	−	2.97227i	0.885736	−	0.464190i
							\(43\)	−0.679214	+	4.28839i	−0.103579	+	0.653974i	0.880202	+	0.474600i	\(0.157407\pi\)
−0.983781	+	0.179374i	\(0.942593\pi\)
\(47\)	−6.75727	+	3.66890i	−0.985649	+	0.535164i	−0.887895	−	0.460047i	\(-0.847833\pi\)
							−0.0977541	+	0.995211i	\(0.531166\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.2.be.a.12.13	✓	832
7.3	odd	6	inner	287.2.be.a.94.14	yes	832
41.24	odd	40	inner	287.2.be.a.229.14	yes	832
287.24	even	120	inner	287.2.be.a.24.13	yes	832

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.2.be.a.12.13	✓	832	1.1	even	1	trivial
287.2.be.a.24.13	yes	832	287.24	even	120	inner
287.2.be.a.94.14	yes	832	7.3	odd	6	inner
287.2.be.a.229.14	yes	832	41.24	odd	40	inner