Embedded newform 287.3.x.a.31.20

• Label: 287.3.x.a.31.20
• 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			31.20
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.113807 + 1.08280i) q^{2} +(-0.534489 + 0.925762i) q^{3} +(2.75308 + 0.585184i) q^{4} +(5.36495 + 4.83062i) q^{5} +(-0.941591 - 0.684106i) q^{6} +(-3.56891 - 6.02187i) q^{7} +(-2.29275 + 7.05637i) q^{8} +(3.92864 + 6.80461i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.113807	+	1.08280i	−0.0569037	+	0.541402i	0.928520	+	0.371283i	\(0.121082\pi\)
							−0.985423	+	0.170119i	\(0.945585\pi\)
\(3\)	−0.534489	+	0.925762i	−0.178163	+	0.308587i	−0.941251	−	0.337707i	\(-0.890349\pi\)
							0.763088	+	0.646294i	\(0.223682\pi\)
\(5\)	5.36495	+	4.83062i	1.07299	+	0.966125i	0.999514	−	0.0311721i	\(-0.00992400\pi\)
							0.0734763	+	0.997297i	\(0.476591\pi\)
\(7\)	−3.56891	−	6.02187i	−0.509844	−	0.860267i
							\(11\)	1.45419	−	1.30936i	0.132199	−	0.119033i	−0.600379	−	0.799716i	\(-0.704984\pi\)
0.732578	+	0.680683i	\(0.238317\pi\)
\(13\)	8.90876	+	6.47259i	0.685289	+	0.497892i	0.875108	−	0.483927i	\(-0.160790\pi\)
							−0.189819	+	0.981819i	\(0.560790\pi\)
\(17\)	−17.1355	−	19.0309i	−1.00797	−	1.11946i	−0.992825	−	0.119579i	\(-0.961846\pi\)
							−0.0151454	−	0.999885i	\(-0.504821\pi\)
\(19\)	15.7643	+	7.01873i	0.829701	+	0.369407i	0.777231	−	0.629216i	\(-0.216624\pi\)
							0.0524704	+	0.998622i	\(0.483290\pi\)
\(23\)	2.24252	−	21.3361i	0.0975008	−	0.927658i	−0.830986	−	0.556293i	\(-0.812223\pi\)
							0.928487	−	0.371365i	\(-0.121110\pi\)
\(29\)	−8.56263	+	2.78217i	−0.295263	+	0.0959367i	−0.452903	−	0.891560i	\(-0.649612\pi\)
							0.157640	+	0.987497i	\(0.449612\pi\)
\(31\)	−11.6082	+	10.4521i	−0.374458	+	0.337163i	−0.834775	−	0.550591i	\(-0.814402\pi\)
							0.460317	+	0.887754i	\(0.347736\pi\)
\(37\)	−48.2230	+	53.5571i	−1.30332	+	1.44749i	−0.482965	+	0.875639i	\(0.660440\pi\)
							−0.820359	+	0.571849i	\(0.806226\pi\)
\(41\)	38.2834	−	14.6760i	0.933740	−	0.357951i
							\(43\)	26.5269	+	19.2729i	0.616905	+	0.448208i	0.851839	−	0.523804i	\(-0.175487\pi\)
−0.234934	+	0.972011i	\(0.575487\pi\)
\(47\)	−0.998491	+	9.50001i	−0.0212445	+	0.202128i	−0.999996	−	0.00292343i	\(-0.999069\pi\)
							0.978751	+	0.205051i	\(0.0657361\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.20	✓	432
7.5	odd	6	inner	287.3.x.a.236.35	yes	432
41.4	even	10	inner	287.3.x.a.45.35	yes	432
287.250	odd	30	inner	287.3.x.a.250.20	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.20	✓	432	1.1	even	1	trivial
287.3.x.a.45.35	yes	432	41.4	even	10	inner
287.3.x.a.236.35	yes	432	7.5	odd	6	inner
287.3.x.a.250.20	yes	432	287.250	odd	30	inner