Embedded newform 490.2.w.a.103.26

• Label: 490.2.w.a.103.26
• 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			103.26
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.593820 - 0.804598i) q^{2} +(2.09713 - 0.396799i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(1.87344 + 1.22075i) q^{5} +(0.926056 - 1.92297i) q^{6} +(1.00239 - 2.44851i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(1.44789 - 0.568256i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	2.09713	−	0.396799i	1.21078	−	0.229092i	0.458958	−	0.888458i	\(-0.348223\pi\)
0.751822	+	0.659366i	\(0.229175\pi\)
\(5\)	1.87344	+	1.22075i	0.837828	+	0.545934i
							\(7\)	1.00239	−	2.44851i	0.378867	−	0.925451i
\(11\)	−0.581144	+	1.48073i	−0.175222	+	0.446457i								−0.991323	−	0.131452i	\(-0.958036\pi\)
							0.816101	+	0.577909i	\(0.196131\pi\)
\(13\)	−1.68739	−	0.190123i	−0.467997	−	0.0527306i	−0.125183	−	0.992134i	\(-0.539952\pi\)
							−0.342815	+	0.939403i	\(0.611380\pi\)
\(17\)	−0.461878	−	0.0172823i	−0.112022	−	0.00419156i	−0.0186785	−	0.999826i	\(-0.505946\pi\)
							−0.0933434	+	0.995634i	\(0.529755\pi\)
\(19\)	−2.81142	+	4.86953i	−0.644985	+	1.11715i	0.339320	+	0.940671i	\(0.389803\pi\)
							−0.984305	+	0.176476i	\(0.943530\pi\)
\(23\)	−0.160054	−	4.27753i	−0.0333735	−	0.891926i	−0.910957	−	0.412502i	\(-0.864655\pi\)
							0.877583	−	0.479424i	\(-0.159155\pi\)
\(29\)	2.65640	−	0.606305i	0.493281	−	0.112588i	0.0313578	−	0.999508i	\(-0.490017\pi\)
							0.461923	+	0.886920i	\(0.347160\pi\)
\(31\)	0.328382	−	0.189592i	0.0589792	−	0.0340517i	−0.470220	−	0.882549i	\(-0.655826\pi\)
							0.529200	+	0.848497i	\(0.322492\pi\)
\(37\)	−1.27160	+	0.672062i	−0.209050	+	0.110486i	−0.568517	−	0.822672i	\(-0.692483\pi\)
							0.359466	+	0.933158i	\(0.382959\pi\)
\(41\)	3.70477	+	7.69303i	0.578587	+	1.20145i	0.960766	+	0.277360i	\(0.0894596\pi\)
							−0.382179	+	0.924088i	\(0.624826\pi\)
\(43\)	−3.23868	−	9.25562i	−0.493894	−	1.41147i	−0.873729	−	0.486414i	\(-0.838305\pi\)
							0.379834	−	0.925054i	\(-0.375981\pi\)
\(47\)	−1.81003	−	1.33586i	−0.264019	−	0.194855i	0.453763	−	0.891122i	\(-0.350081\pi\)
							−0.717783	+	0.696267i	\(0.754843\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.26	✓	672
5.2	odd	4	inner	490.2.w.a.397.17	yes	672
49.10	odd	42	inner	490.2.w.a.353.17	yes	672
245.157	even	84	inner	490.2.w.a.157.26	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.26	✓	672	1.1	even	1	trivial
490.2.w.a.157.26	yes	672	245.157	even	84	inner
490.2.w.a.353.17	yes	672	49.10	odd	42	inner
490.2.w.a.397.17	yes	672	5.2	odd	4	inner