Embedded newform 287.3.k.a.124.14

• Label: 287.3.k.a.124.14
• 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			124.14
Character	\(\chi\)	\(=\)	287.124
Dual form			287.3.k.a.206.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09053 + 1.88885i) q^{2} +(1.31239 - 0.757706i) q^{3} +(-0.378493 - 0.655569i) q^{4} +(-2.19911 - 1.26966i) q^{5} +3.30519i q^{6} +(1.49325 - 6.83887i) q^{7} -7.07318 q^{8} +(-3.35176 + 5.80542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 108 q - 2 q^{2} - 6 q^{3} - 106 q^{4} - 6 q^{5} - 4 q^{7} - 4 q^{8} + 164 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09053	+	1.88885i	−0.545263	+	0.944423i	0.453327	+	0.891344i	\(0.350237\pi\)
							−0.998590	+	0.0530789i	\(0.983097\pi\)
\(3\)	1.31239	−	0.757706i	0.437462	−	0.252569i	−0.265059	−	0.964232i	\(-0.585391\pi\)
							0.702520	+	0.711664i	\(0.252058\pi\)
\(5\)	−2.19911	−	1.26966i	−0.439822	−	0.253931i	0.263700	−	0.964605i	\(-0.415057\pi\)
							−0.703522	+	0.710673i	\(0.748390\pi\)
\(7\)	1.49325	−	6.83887i	0.213321	−	0.976982i
							\(11\)	−1.86781	−	3.23514i	−0.169801	−	0.294104i	0.768549	−	0.639791i	\(-0.220979\pi\)
−0.938350	+	0.345687i	\(0.887646\pi\)
\(13\)		−	18.1566i		−	1.39666i	−0.715776	−	0.698330i	\(-0.753927\pi\)
							0.715776	−	0.698330i	\(-0.246073\pi\)
\(17\)	−21.9598	+	12.6785i	−1.29175	+	0.745795i	−0.978965	−	0.204027i	\(-0.934597\pi\)
							−0.312790	+	0.949822i	\(0.601264\pi\)
\(19\)	−12.1620	−	7.02173i	−0.640105	−	0.369565i	0.144550	−	0.989498i	\(-0.453827\pi\)
							−0.784655	+	0.619933i	\(0.787160\pi\)
\(23\)	11.7410	−	20.3360i	0.510479	−	0.884175i	−0.489448	−	0.872033i	\(-0.662802\pi\)
							0.999926	−	0.0121423i	\(-0.00386512\pi\)
\(29\)	6.48213			0.223522			0.111761	−	0.993735i	\(-0.464351\pi\)
							0.111761	+	0.993735i	\(0.464351\pi\)
\(31\)	−37.7628	+	21.8024i	−1.21815	+	0.703302i	−0.964523	−	0.263999i	\(-0.914959\pi\)
							−0.253632	+	0.967301i	\(0.581625\pi\)
\(37\)	34.1489	−	59.1477i	0.922944	−	1.59859i	0.128110	−	0.991760i	\(-0.459109\pi\)
							0.794834	−	0.606827i	\(-0.207558\pi\)
\(41\)		−	6.40312i		−	0.156174i
							\(43\)	14.9564			0.347823			0.173912	−	0.984761i	\(-0.444359\pi\)
0.173912	+	0.984761i	\(0.444359\pi\)
\(47\)	48.5451	+	28.0275i	1.03288	+	0.596331i	0.917807	−	0.397027i	\(-0.129958\pi\)
							0.115068	+	0.993358i	\(0.463291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.k.a.124.14	✓	108
7.3	odd	6	inner	287.3.k.a.206.14	yes	108

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.k.a.124.14	✓	108	1.1	even	1	trivial
287.3.k.a.206.14	yes	108	7.3	odd	6	inner