Embedded newform 287.3.i.a.40.20

• Label: 287.3.i.a.40.20
• Level: $287$
• Weight: $3$
• Character: 287.40
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(108\)
Relative dimension:	\(54\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			40.20
Character	\(\chi\)	\(=\)	287.40
Dual form			287.3.i.a.122.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.702038 + 1.21597i) q^{2} +(2.20646 + 3.82171i) q^{3} +(1.01428 + 1.75679i) q^{4} +(3.32560 + 1.92003i) q^{5} -6.19609 q^{6} +(5.26852 - 4.60898i) q^{7} -8.46457 q^{8} +(-5.23697 + 9.07071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 106 q^{4} - 6 q^{5} + 20 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.702038	+	1.21597i	−0.351019	+	0.607983i	−0.986428	−	0.164193i	\(-0.947498\pi\)
							0.635409	+	0.772176i	\(0.280831\pi\)
\(3\)	2.20646	+	3.82171i	0.735488	+	1.27390i	0.954509	+	0.298183i	\(0.0963805\pi\)
							−0.219020	+	0.975720i	\(0.570286\pi\)
\(5\)	3.32560	+	1.92003i	0.665120	+	0.384007i	0.794225	−	0.607624i	\(-0.207877\pi\)
							−0.129105	+	0.991631i	\(0.541211\pi\)
\(7\)	5.26852	−	4.60898i	0.752646	−	0.658426i
							\(11\)	−16.8836	+	9.74776i	−1.53487	+	0.886160i	−0.535747	+	0.844378i	\(0.679970\pi\)
−0.999127	+	0.0417816i	\(0.986697\pi\)
\(13\)	18.3297			1.40998			0.704990	−	0.709217i	\(-0.250951\pi\)
							0.704990	+	0.709217i	\(0.250951\pi\)
\(17\)	2.36374	+	4.09413i	0.139044	+	0.240831i	0.927135	−	0.374728i	\(-0.122264\pi\)
							−0.788091	+	0.615559i	\(0.788930\pi\)
\(19\)	12.9142	−	22.3680i	0.679693	−	1.17726i	−0.295380	−	0.955380i	\(-0.595446\pi\)
							0.975073	−	0.221883i	\(-0.0712203\pi\)
\(23\)	−14.5906	+	25.2716i	−0.634372	+	1.09876i	0.352276	+	0.935896i	\(0.385408\pi\)
							−0.986648	+	0.162868i	\(0.947925\pi\)
\(29\)		−	40.8277i		−	1.40785i	−0.710274	−	0.703925i	\(-0.751429\pi\)
							0.710274	−	0.703925i	\(-0.248571\pi\)
\(31\)	5.93826	−	3.42846i	0.191557	−	0.110595i	−0.401154	−	0.916010i	\(-0.631391\pi\)
							0.592711	+	0.805415i	\(0.298058\pi\)
\(37\)	−28.7405	+	49.7800i	−0.776771	+	1.34541i	0.157023	+	0.987595i	\(0.449810\pi\)
							−0.933794	+	0.357811i	\(0.883523\pi\)
\(41\)	40.9965	−	0.533559i	0.999915	−	0.0130136i
							\(43\)	27.8509			0.647696			0.323848	−	0.946109i	\(-0.395023\pi\)
0.323848	+	0.946109i	\(0.395023\pi\)
\(47\)	9.83269	−	17.0307i	0.209206	−	0.362356i	−0.742259	−	0.670114i	\(-0.766245\pi\)
							0.951465	+	0.307758i	\(0.0995787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.i.a.40.20	yes	108
7.3	odd	6	inner	287.3.i.a.122.19	yes	108
41.40	even	2	inner	287.3.i.a.40.19	✓	108
287.122	odd	6	inner	287.3.i.a.122.20	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.i.a.40.19	✓	108	41.40	even	2	inner
287.3.i.a.40.20	yes	108	1.1	even	1	trivial
287.3.i.a.122.19	yes	108	7.3	odd	6	inner
287.3.i.a.122.20	yes	108	287.122	odd	6	inner