Embedded newform 287.3.x.a.31.4

• Label: 287.3.x.a.31.4
• 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			31.4
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.374024 + 3.55860i) q^{2} +(0.0587614 - 0.101778i) q^{3} +(-8.61118 - 1.83036i) q^{4} +(2.71663 + 2.44606i) q^{5} +(0.340208 + 0.247176i) q^{6} +(2.46506 + 6.55160i) q^{7} +(5.31143 - 16.3469i) q^{8} +(4.49309 + 7.78227i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.374024	+	3.55860i	−0.187012	+	1.77930i	0.351010	+	0.936372i	\(0.385838\pi\)
							−0.538022	+	0.842931i	\(0.680828\pi\)
\(3\)	0.0587614	−	0.101778i	0.0195871	−	0.0339259i	−0.856066	−	0.516867i	\(-0.827098\pi\)
							0.875653	+	0.482941i	\(0.160431\pi\)
\(5\)	2.71663	+	2.44606i	0.543326	+	0.489213i	0.894482	−	0.447104i	\(-0.147545\pi\)
							−0.351156	+	0.936317i	\(0.614211\pi\)
\(7\)	2.46506	+	6.55160i	0.352152	+	0.935943i
							\(11\)	−7.20460	+	6.48705i	−0.654963	+	0.589732i	−0.928140	−	0.372232i	\(-0.878592\pi\)
0.273176	+	0.961964i	\(0.411926\pi\)
\(13\)	−1.99679	−	1.45076i	−0.153600	−	0.111597i	0.508331	−	0.861162i	\(-0.330263\pi\)
							−0.661931	+	0.749565i	\(0.730263\pi\)
\(17\)	−3.30592	−	3.67160i	−0.194466	−	0.215977i	0.638024	−	0.770017i	\(-0.279752\pi\)
							−0.832490	+	0.554040i	\(0.813085\pi\)
\(19\)	4.61620	+	2.05526i	0.242958	+	0.108172i	0.524603	−	0.851347i	\(-0.324214\pi\)
							−0.281646	+	0.959518i	\(0.590880\pi\)
\(23\)	1.42448	−	13.5530i	0.0619340	−	0.589262i	−0.918910	−	0.394467i	\(-0.870929\pi\)
							0.980844	−	0.194795i	\(-0.0624042\pi\)
\(29\)	−39.6082	+	12.8695i	−1.36580	+	0.443775i	−0.897975	−	0.440046i	\(-0.854962\pi\)
							−0.467825	+	0.883821i	\(0.654962\pi\)
\(31\)	−2.28110	+	2.05391i	−0.0735839	+	0.0662553i	−0.705102	−	0.709106i	\(-0.749099\pi\)
							0.631518	+	0.775361i	\(0.282432\pi\)
\(37\)	11.6741	−	12.9654i	0.315516	−	0.350416i	−0.564438	−	0.825476i	\(-0.690907\pi\)
							0.879954	+	0.475060i	\(0.157574\pi\)
\(41\)	36.7460	+	18.1861i	0.896243	+	0.443563i
							\(43\)	2.08126	+	1.51212i	0.0484013	+	0.0351656i	0.611723	−	0.791072i	\(-0.290477\pi\)
−0.563322	+	0.826238i	\(0.690477\pi\)
\(47\)	−7.91300	+	75.2871i	−0.168362	+	1.60185i	0.505387	+	0.862893i	\(0.331350\pi\)
							−0.673748	+	0.738961i	\(0.735317\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.4	✓	432
7.5	odd	6	inner	287.3.x.a.236.51	yes	432
41.4	even	10	inner	287.3.x.a.45.51	yes	432
287.250	odd	30	inner	287.3.x.a.250.4	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.4	✓	432	1.1	even	1	trivial
287.3.x.a.45.51	yes	432	41.4	even	10	inner
287.3.x.a.236.51	yes	432	7.5	odd	6	inner
287.3.x.a.250.4	yes	432	287.250	odd	30	inner