Embedded newform 287.3.x.a.31.1

• Label: 287.3.x.a.31.1
• 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			31.1
Character	\(\chi\)	\(=\)	287.31
Dual form			287.3.x.a.250.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.404206 + 3.84576i) q^{2} +(-2.18229 + 3.77983i) q^{3} +(-10.7139 - 2.27731i) q^{4} +(6.18506 + 5.56906i) q^{5} +(-13.6543 - 9.92039i) q^{6} +(-5.32339 - 4.54549i) q^{7} +(8.30884 - 25.5720i) q^{8} +(-5.02476 - 8.70314i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 432 q - 3 q^{2} + 101 q^{4} - 9 q^{5} - 10 q^{7} - 40 q^{8} - 624 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.404206	+	3.84576i	−0.202103	+	1.92288i	0.153321	+	0.988176i	\(0.451003\pi\)
							−0.355424	+	0.934705i	\(0.615663\pi\)
\(3\)	−2.18229	+	3.77983i	−0.727429	+	1.25994i	0.230537	+	0.973064i	\(0.425952\pi\)
							−0.957966	+	0.286881i	\(0.907382\pi\)
\(5\)	6.18506	+	5.56906i	1.23701	+	1.11381i	0.989440	+	0.144942i	\(0.0462995\pi\)
							0.247572	+	0.968869i	\(0.420367\pi\)
\(7\)	−5.32339	−	4.54549i	−0.760485	−	0.649356i
							\(11\)	5.77639	−	5.20109i	0.525126	−	0.472826i	−0.363412	−	0.931629i	\(-0.618388\pi\)
0.888538	+	0.458803i	\(0.151722\pi\)
\(13\)	−0.445099	−	0.323384i	−0.0342384	−	0.0248757i	0.570534	−	0.821274i	\(-0.306736\pi\)
							−0.604773	+	0.796398i	\(0.706736\pi\)
\(17\)	−7.10872	−	7.89503i	−0.418160	−	0.464413i	0.496855	−	0.867833i	\(-0.334488\pi\)
							−0.915015	+	0.403420i	\(0.867821\pi\)
\(19\)	−33.8844	−	15.0863i	−1.78339	−	0.794017i	−0.980121	−	0.198402i	\(-0.936425\pi\)
							−0.803270	−	0.595615i	\(-0.796908\pi\)
\(23\)	−1.32351	+	12.5923i	−0.0575439	+	0.547493i	0.927333	+	0.374237i	\(0.122095\pi\)
							−0.984877	+	0.173256i	\(0.944571\pi\)
\(29\)	−7.20818	+	2.34208i	−0.248558	+	0.0807614i	−0.430646	−	0.902521i	\(-0.641714\pi\)
							0.182088	+	0.983282i	\(0.441714\pi\)
\(31\)	−38.2501	+	34.4406i	−1.23388	+	1.11099i	−0.243892	+	0.969802i	\(0.578424\pi\)
							−0.989983	+	0.141184i	\(0.954909\pi\)
\(37\)	13.2243	−	14.6871i	0.357415	−	0.396949i	−0.537443	−	0.843300i	\(-0.680610\pi\)
							0.894858	+	0.446350i	\(0.147277\pi\)
\(41\)	−17.1512	+	37.2402i	−0.418323	+	0.908298i
							\(43\)	−11.5855	−	8.41733i	−0.269429	−	0.195752i	0.444864	−	0.895598i	\(-0.353252\pi\)
−0.714294	+	0.699846i	\(0.753252\pi\)
\(47\)	2.45409	−	23.3491i	0.0522147	−	0.496790i	−0.936895	−	0.349611i	\(-0.886314\pi\)
							0.989110	−	0.147179i	\(-0.0470195\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	287.3.x.a.31.1	✓	432
7.5	odd	6	inner	287.3.x.a.236.54	yes	432
41.4	even	10	inner	287.3.x.a.45.54	yes	432
287.250	odd	30	inner	287.3.x.a.250.1	yes	432

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
287.3.x.a.31.1	✓	432	1.1	even	1	trivial
287.3.x.a.45.54	yes	432	41.4	even	10	inner
287.3.x.a.236.54	yes	432	7.5	odd	6	inner
287.3.x.a.250.1	yes	432	287.250	odd	30	inner