Embedded newform 287.3.x.a.31.10

• Label: 287.3.x.a.31.10
• Level: $287$
• Weight: $3$
• Character: 287.31
• 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.x (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.82018358714\)
Analytic rank:	\(0\)
Dimension:	\(432\)
Relative dimension:	\(54\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			31.10
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.277470 + 2.63995i) q^{2} +(-0.490949 + 0.850348i) q^{3} +(-2.97975 - 0.633365i) q^{4} +(3.54226 + 3.18946i) q^{5} +(-2.10865 - 1.53203i) q^{6} +(6.74819 - 1.86061i) q^{7} +(-0.782288 + 2.40763i) q^{8} +(4.01794 + 6.95927i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(206\)	\(211\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{7}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.277470	+	2.63995i	−0.138735	+	1.31997i	0.674602	+	0.738182i	\(0.264315\pi\)
							−0.813337	+	0.581793i	\(0.802351\pi\)
\(3\)	−0.490949	+	0.850348i	−0.163650	+	0.283449i	−0.936175	−	0.351535i	\(-0.885660\pi\)
							0.772525	+	0.634984i	\(0.218993\pi\)
\(5\)	3.54226	+	3.18946i	0.708452	+	0.637893i	0.942451	−	0.334344i	\(-0.108515\pi\)
							−0.233999	+	0.972237i	\(0.575181\pi\)
\(7\)	6.74819	−	1.86061i	0.964028	−	0.265802i
							\(11\)	14.6584	−	13.1985i	1.33258	−	1.19986i	0.369946	−	0.929053i	\(-0.379376\pi\)
0.962634	−	0.270807i	\(-0.0872905\pi\)
\(13\)	5.63587	+	4.09470i	0.433528	+	0.314977i	0.783058	−	0.621948i	\(-0.213659\pi\)
							−0.349530	+	0.936925i	\(0.613659\pi\)
\(17\)	17.7080	+	19.6668i	1.04165	+	1.15687i	0.987384	+	0.158343i	\(0.0506150\pi\)
							0.0542654	+	0.998527i	\(0.482718\pi\)
\(19\)	−24.7005	−	10.9974i	−1.30003	−	0.578810i	−0.364217	−	0.931314i	\(-0.618663\pi\)
							−0.935810	+	0.352504i	\(0.885330\pi\)
\(23\)	1.13952	−	10.8418i	0.0495445	−	0.471384i	−0.941418	−	0.337243i	\(-0.890506\pi\)
							0.990962	−	0.134141i	\(-0.0428277\pi\)
\(29\)	−33.3402	+	10.8329i	−1.14966	+	0.373548i	−0.821016	−	0.570905i	\(-0.806593\pi\)
							−0.328647	+	0.944453i	\(0.606593\pi\)
\(31\)	29.3031	−	26.3846i	0.945260	−	0.851116i	−0.0437381	−	0.999043i	\(-0.513927\pi\)
							0.988998	+	0.147927i	\(0.0472601\pi\)
\(37\)	−34.2206	+	38.0058i	−0.924881	+	1.02718i	0.0746699	+	0.997208i	\(0.476210\pi\)
							−0.999551	+	0.0299758i	\(0.990457\pi\)
\(41\)	−39.6594	−	10.3985i	−0.967303	−	0.253623i
							\(43\)	−39.2813	−	28.5395i	−0.913518	−	0.663710i	0.0283839	−	0.999597i	\(-0.490964\pi\)
−0.941902	+	0.335887i	\(0.890964\pi\)
\(47\)	1.64467	−	15.6480i	0.0349930	−	0.332936i	−0.962994	−	0.269522i	\(-0.913134\pi\)
							0.997987	−	0.0634144i	\(-0.0201990\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.10	✓	432
7.5	odd	6	inner	287.3.x.a.236.45	yes	432
41.4	even	10	inner	287.3.x.a.45.45	yes	432
287.250	odd	30	inner	287.3.x.a.250.10	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.10	✓	432	1.1	even	1	trivial
287.3.x.a.45.45	yes	432	41.4	even	10	inner
287.3.x.a.236.45	yes	432	7.5	odd	6	inner
287.3.x.a.250.10	yes	432	287.250	odd	30	inner