Embedded newform 490.2.w.a.103.9

• Label: 490.2.w.a.103.9
• Level: $490$
• Weight: $2$
• Character: 490.103
• Analytic conductor: $3.913$
• Analytic rank: $0$
• Dimension: $672$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	490.w (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.91266969904\)
Analytic rank:	\(0\)
Dimension:	\(672\)
Relative dimension:	\(28\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			103.9
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.593820 + 0.804598i) q^{2} +(0.858720 - 0.162479i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(2.23532 + 0.0576957i) q^{5} +(-0.379195 + 0.787408i) q^{6} +(2.57637 - 0.601908i) q^{7} +(0.943883 + 0.330279i) q^{8} +(-2.08162 + 0.816976i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{29}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.593820	+	0.804598i	−0.419894	+	0.568937i
							\(3\)	0.858720	−	0.162479i	0.495782	−	0.0938071i	0.0679935	−	0.997686i	\(-0.478340\pi\)
0.427789	+	0.903879i	\(0.359293\pi\)
\(5\)	2.23532	+	0.0576957i	0.999667	+	0.0258023i
							\(7\)	2.57637	−	0.601908i	0.973778	−	0.227500i
\(11\)	−2.11322	+	5.38440i	−0.637160	+	1.62346i								0.137077	+	0.990560i	\(0.456229\pi\)
							−0.774236	+	0.632896i	\(0.781866\pi\)
\(13\)	0.808550	+	0.0911017i	0.224252	+	0.0252671i	0.223375	−	0.974733i	\(-0.428292\pi\)
							0.000876313		1.00000i	\(0.499721\pi\)
\(17\)	3.43010	+	0.128345i	0.831922	+	0.0311283i	0.449959	−	0.893049i	\(-0.351439\pi\)
							0.381963	+	0.924177i	\(0.375248\pi\)
\(19\)	3.76383	−	6.51914i	0.863482	−	1.49559i	−0.00506512	−	0.999987i	\(-0.501612\pi\)
							0.868547	−	0.495607i	\(-0.165054\pi\)
\(23\)	−0.0216924	−	0.579741i	−0.00452317	−	0.120884i	−0.999733	−	0.0231194i	\(-0.992640\pi\)
							0.995210	−	0.0977649i	\(-0.0311693\pi\)
\(29\)	−0.638402	+	0.145711i	−0.118548	+	0.0270579i	−0.281383	−	0.959595i	\(-0.590793\pi\)
							0.162835	+	0.986653i	\(0.447936\pi\)
\(31\)	4.49880	−	2.59739i	0.808009	−	0.466504i	−0.0382549	−	0.999268i	\(-0.512180\pi\)
							0.846264	+	0.532764i	\(0.178847\pi\)
\(37\)	−8.82463	+	4.66396i	−1.45076	+	0.766750i	−0.992520	−	0.122079i	\(-0.961044\pi\)
							−0.458241	+	0.888828i	\(0.651520\pi\)
\(41\)	1.30896	+	2.71808i	0.204425	+	0.424493i	0.977825	−	0.209426i	\(-0.0671595\pi\)
							−0.773399	+	0.633919i	\(0.781445\pi\)
\(43\)	3.04919	+	8.71408i	0.464997	+	1.32889i	0.902895	+	0.429861i	\(0.141437\pi\)
							−0.437898	+	0.899025i	\(0.644277\pi\)
\(47\)	−7.03388	−	5.19124i	−1.02600	−	0.757220i	−0.0555305	−	0.998457i	\(-0.517685\pi\)
							−0.970466	+	0.241237i	\(0.922447\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.9	✓	672
5.2	odd	4	inner	490.2.w.a.397.6	yes	672
49.10	odd	42	inner	490.2.w.a.353.6	yes	672
245.157	even	84	inner	490.2.w.a.157.9	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.9	✓	672	1.1	even	1	trivial
490.2.w.a.157.9	yes	672	245.157	even	84	inner
490.2.w.a.353.6	yes	672	49.10	odd	42	inner
490.2.w.a.397.6	yes	672	5.2	odd	4	inner