Embedded newform 490.2.w.a.103.21

• Label: 490.2.w.a.103.21
• 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			103.21
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.593820 - 0.804598i) q^{2} +(-0.561836 + 0.106305i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(-2.22607 + 0.211211i) q^{5} +(-0.248097 + 0.515178i) q^{6} +(1.35289 - 2.27370i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(-2.48826 + 0.976571i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−0.561836	+	0.106305i	−0.324376	+	0.0613753i	−0.345550	−	0.938401i	\(-0.612307\pi\)
0.0211732	+	0.999776i	\(0.493260\pi\)
\(5\)	−2.22607	+	0.211211i	−0.995529	+	0.0944566i
							\(7\)	1.35289	−	2.27370i	0.511343	−	0.859377i
\(11\)	−1.52442	+	3.88416i	−0.459630	+	1.17112i								0.493820	+	0.869564i	\(0.335600\pi\)
							−0.953449	+	0.301553i	\(0.902495\pi\)
\(13\)	−3.96616	−	0.446879i	−1.10002	−	0.123942i	−0.456757	−	0.889591i	\(-0.650989\pi\)
							−0.643258	+	0.765649i	\(0.722418\pi\)
\(17\)	−3.04472	−	0.113925i	−0.738453	−	0.0276310i	−0.334481	−	0.942402i	\(-0.608561\pi\)
							−0.403972	+	0.914771i	\(0.632371\pi\)
\(19\)	1.51403	−	2.62238i	0.347342	−	0.601614i	−0.638434	−	0.769676i	\(-0.720418\pi\)
							0.985776	+	0.168062i	\(0.0537509\pi\)
\(23\)	0.0241896	+	0.646479i	0.00504387	+	0.134800i	0.999547	+	0.0300941i	\(0.00958069\pi\)
							−0.994503	+	0.104706i	\(0.966610\pi\)
\(29\)	−5.02986	+	1.14803i	−0.934022	+	0.213185i	−0.662341	−	0.749203i	\(-0.730437\pi\)
							−0.271682	+	0.962387i	\(0.587580\pi\)
\(31\)	−6.81876	+	3.93681i	−1.22469	+	0.707072i	−0.965913	−	0.258866i	\(-0.916651\pi\)
							−0.258772	+	0.965938i	\(0.583318\pi\)
\(37\)	−1.13591	+	0.600346i	−0.186743	+	0.0986963i	−0.558027	−	0.829823i	\(-0.688441\pi\)
							0.371284	+	0.928519i	\(0.378918\pi\)
\(41\)	−0.746180	−	1.54946i	−0.116534	−	0.241985i	0.834541	−	0.550946i	\(-0.185733\pi\)
							−0.951075	+	0.308961i	\(0.900019\pi\)
\(43\)	−0.282781	−	0.808143i	−0.0431238	−	0.123241i	0.920308	−	0.391195i	\(-0.127938\pi\)
							−0.963432	+	0.267954i	\(0.913652\pi\)
\(47\)	−10.4495	−	7.71209i	−1.52422	−	1.12492i	−0.952527	−	0.304454i	\(-0.901526\pi\)
							−0.571690	−	0.820470i	\(-0.693712\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.21	✓	672
5.2	odd	4	inner	490.2.w.a.397.22	yes	672
49.10	odd	42	inner	490.2.w.a.353.22	yes	672
245.157	even	84	inner	490.2.w.a.157.21	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.21	✓	672	1.1	even	1	trivial
490.2.w.a.157.21	yes	672	245.157	even	84	inner
490.2.w.a.353.22	yes	672	49.10	odd	42	inner
490.2.w.a.397.22	yes	672	5.2	odd	4	inner