Embedded newform 287.3.k.a.124.16

• Label: 287.3.k.a.124.16
• Level: $287$
• Weight: $3$
• Character: 287.124
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $108$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.k (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			124.16
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.981202 + 1.69949i) q^{2} +(-4.51577 + 2.60718i) q^{3} +(0.0744841 + 0.129010i) q^{4} +(0.831105 + 0.479839i) q^{5} -10.2327i q^{6} +(6.78912 + 1.70524i) q^{7} -8.14195 q^{8} +(9.09481 - 15.7527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.981202	+	1.69949i	−0.490601	+	0.849746i	−0.999941	−	0.0108191i	\(-0.996556\pi\)
							0.509340	+	0.860565i	\(0.329889\pi\)
\(3\)	−4.51577	+	2.60718i	−1.50526	+	0.869061i	−0.505276	+	0.862958i	\(0.668609\pi\)
							−0.999981	+	0.00610341i	\(0.998057\pi\)
\(5\)	0.831105	+	0.479839i	0.166221	+	0.0959677i	0.580803	−	0.814044i	\(-0.302739\pi\)
							−0.414582	+	0.910012i	\(0.636072\pi\)
\(7\)	6.78912	+	1.70524i	0.969874	+	0.243605i
							\(11\)	−7.08816	−	12.2770i	−0.644378	−	1.11610i	−0.984445	−	0.175694i	\(-0.943783\pi\)
0.340067	−	0.940401i	\(-0.389550\pi\)
\(13\)		−	24.7886i		−	1.90682i	−0.301680	−	0.953409i	\(-0.597548\pi\)
							0.301680	−	0.953409i	\(-0.402452\pi\)
\(17\)	−15.3132	+	8.84110i	−0.900778	+	0.520064i	−0.877453	−	0.479664i	\(-0.840759\pi\)
							−0.0233255	+	0.999728i	\(0.507425\pi\)
\(19\)	24.7381	+	14.2826i	1.30201	+	0.751714i	0.980748	−	0.195278i	\(-0.0625608\pi\)
							0.321259	+	0.946992i	\(0.395894\pi\)
\(23\)	8.32677	−	14.4224i	0.362034	−	0.627061i	−0.626262	−	0.779613i	\(-0.715416\pi\)
							0.988295	+	0.152552i	\(0.0487492\pi\)
\(29\)	−7.15581			−0.246752			−0.123376	−	0.992360i	\(-0.539372\pi\)
							−0.123376	+	0.992360i	\(0.539372\pi\)
\(31\)	24.1572	−	13.9472i	0.779264	−	0.449908i	−0.0569053	−	0.998380i	\(-0.518123\pi\)
							0.836169	+	0.548471i	\(0.184790\pi\)
\(37\)	−17.7089	+	30.6726i	−0.478618	+	0.828990i	−0.999699	−	0.0245165i	\(-0.992195\pi\)
							0.521082	+	0.853507i	\(0.325529\pi\)
\(41\)			6.40312i			0.156174i
							\(43\)	−66.8085			−1.55369			−0.776844	−	0.629694i	\(-0.783180\pi\)
−0.776844	+	0.629694i	\(0.783180\pi\)
\(47\)	−22.6654	−	13.0859i	−0.482242	−	0.278423i	0.239108	−	0.970993i	\(-0.423145\pi\)
							−0.721350	+	0.692570i	\(0.756478\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.16	✓	108
7.3	odd	6	inner	287.3.k.a.206.16	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.16	✓	108	1.1	even	1	trivial
287.3.k.a.206.16	yes	108	7.3	odd	6	inner