Embedded newform 287.3.t.a.20.13

• Label: 287.3.t.a.20.13
• Level: $287$
• Weight: $3$
• Character: 287.20
• Analytic conductor: $7.820$
• Analytic rank: $0$
• Dimension: $432$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 287 = 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	287.t (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label			20.13
Character	\(\chi\)	\(=\)	287.20
Dual form			287.3.t.a.244.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.36022 - 0.766883i) q^{2} +(-3.15575 - 3.15575i) q^{3} +(1.74648 + 1.26889i) q^{4} +(-6.16033 - 4.47574i) q^{5} +(5.02818 + 9.86836i) q^{6} +(6.99999 + 0.00883669i) q^{7} +(2.68581 + 3.69670i) q^{8} +10.9175i q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 20 q^{2} + 196 q^{4} - 8 q^{7} - 20 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{17}{20}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.36022	−	0.766883i	−1.18011	−	0.383442i	−0.347704	−	0.937605i	\(-0.613038\pi\)
							−0.832408	+	0.554163i	\(0.813038\pi\)
\(3\)	−3.15575	−	3.15575i	−1.05192	−	1.05192i	−0.998576	−	0.0533392i	\(-0.983014\pi\)
							−0.0533392	−	0.998576i	\(-0.516986\pi\)
\(5\)	−6.16033	−	4.47574i	−1.23207	−	0.895148i	−0.235023	−	0.971990i	\(-0.575516\pi\)
							−0.997043	+	0.0768417i	\(0.975516\pi\)
\(7\)	6.99999	+	0.00883669i	0.999999	+	0.00126238i
							\(11\)	−1.58609	+	10.0142i	−0.144190	+	0.910382i	0.804449	+	0.594022i	\(0.202461\pi\)
−0.948639	+	0.316360i	\(0.897539\pi\)
\(13\)	−10.1412	+	5.16718i	−0.780090	+	0.397476i	−0.798225	−	0.602360i	\(-0.794227\pi\)
							0.0181347	+	0.999836i	\(0.494227\pi\)
\(17\)	−1.09196	+	6.89438i	−0.0642331	+	0.405552i	0.934532	+	0.355878i	\(0.115818\pi\)
							−0.998765	+	0.0496738i	\(0.984182\pi\)
\(19\)	22.7713	+	11.6026i	1.19849	+	0.610662i	0.935223	−	0.354060i	\(-0.115199\pi\)
							0.263269	+	0.964722i	\(0.415199\pi\)
\(23\)	9.74155	−	29.9814i	0.423546	−	1.30354i	−0.480834	−	0.876812i	\(-0.659666\pi\)
							0.904380	−	0.426728i	\(-0.140334\pi\)
\(29\)	−28.4756	+	4.51010i	−0.981918	+	0.155521i	−0.626698	−	0.779262i	\(-0.715594\pi\)
							−0.355220	+	0.934783i	\(0.615594\pi\)
\(31\)	10.9917	+	15.1288i	0.354571	+	0.488025i	0.948626	−	0.316399i	\(-0.102474\pi\)
							−0.594055	+	0.804424i	\(0.702474\pi\)
\(37\)	−21.7302	−	15.7879i	−0.587302	−	0.426700i	0.254047	−	0.967192i	\(-0.418238\pi\)
							−0.841349	+	0.540492i	\(0.818238\pi\)
\(41\)	−8.46133	+	40.1174i	−0.206374	+	0.978473i
							\(43\)	61.4153	+	19.9550i	1.42826	+	0.464070i	0.918217	−	0.396077i	\(-0.129629\pi\)
0.510045	+	0.860148i	\(0.329629\pi\)
\(47\)	−21.2658	−	41.7365i	−0.452464	−	0.888010i	−0.998729	−	0.0503977i	\(-0.983951\pi\)
							0.546266	−	0.837612i	\(-0.316049\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.t.a.20.13	✓	432
7.6	odd	2	inner	287.3.t.a.20.14	yes	432
41.39	even	20	inner	287.3.t.a.244.14	yes	432
287.244	odd	20	inner	287.3.t.a.244.13	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.t.a.20.13	✓	432	1.1	even	1	trivial
287.3.t.a.20.14	yes	432	7.6	odd	2	inner
287.3.t.a.244.13	yes	432	287.244	odd	20	inner
287.3.t.a.244.14	yes	432	41.39	even	20	inner