Embedded newform 490.2.w.a.353.17

• Label: 490.2.w.a.353.17
• Level: $490$
• Weight: $2$
• Character: 490.353
• 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			353.17
Character	\(\chi\)	\(=\)	490.353
Dual form			490.2.w.a.397.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.804598 - 0.593820i) q^{2} +(-0.396799 + 2.09713i) q^{3} +(0.294755 - 0.955573i) q^{4} +(2.03446 + 0.927890i) q^{5} +(0.926056 + 1.92297i) q^{6} +(2.44851 - 1.00239i) q^{7} +(-0.330279 - 0.943883i) q^{8} +(-1.44789 - 0.568256i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{13}{42}\right)\)	\(e\left(\frac{3}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.804598	−	0.593820i	0.568937	−	0.419894i
							\(3\)	−0.396799	+	2.09713i	−0.229092	+	1.21078i	0.659366	+	0.751822i	\(0.270825\pi\)
−0.888458	+	0.458958i	\(0.848223\pi\)
\(5\)	2.03446	+	0.927890i	0.909837	+	0.414965i
							\(7\)	2.44851	−	1.00239i	0.925451	−	0.378867i
\(11\)	−0.581144	−	1.48073i	−0.175222	−	0.446457i								0.816101	−	0.577909i	\(-0.196131\pi\)
							−0.991323	+	0.131452i	\(0.958036\pi\)
\(13\)	−0.190123	−	1.68739i	−0.0527306	−	0.467997i	−0.992134	−	0.125183i	\(-0.960048\pi\)
							0.939403	−	0.342815i	\(-0.111380\pi\)
\(17\)	0.0172823	+	0.461878i	0.00419156	+	0.112022i	0.999826	+	0.0186785i	\(0.00594591\pi\)
							−0.995634	+	0.0933434i	\(0.970245\pi\)
\(19\)	2.81142	+	4.86953i	0.644985	+	1.11715i	0.984305	+	0.176476i	\(0.0564697\pi\)
							−0.339320	+	0.940671i	\(0.610197\pi\)
\(23\)	−4.27753	−	0.160054i	−0.891926	−	0.0333735i	−0.412502	−	0.910957i	\(-0.635345\pi\)
							−0.479424	+	0.877583i	\(0.659155\pi\)
\(29\)	−2.65640	−	0.606305i	−0.493281	−	0.112588i	−0.0313578	−	0.999508i	\(-0.509983\pi\)
							−0.461923	+	0.886920i	\(0.652840\pi\)
\(31\)	0.328382	+	0.189592i	0.0589792	+	0.0340517i	0.529200	−	0.848497i	\(-0.322492\pi\)
							−0.470220	+	0.882549i	\(0.655826\pi\)
\(37\)	−0.672062	+	1.27160i	−0.110486	+	0.209050i	−0.933158	−	0.359466i	\(-0.882959\pi\)
							0.822672	+	0.568517i	\(0.192483\pi\)
\(41\)	3.70477	−	7.69303i	0.578587	−	1.20145i	−0.382179	−	0.924088i	\(-0.624826\pi\)
							0.960766	−	0.277360i	\(-0.0894596\pi\)
\(43\)	−9.25562	−	3.23868i	−1.41147	−	0.493894i	−0.486414	−	0.873729i	\(-0.661695\pi\)
							−0.925054	+	0.379834i	\(0.875981\pi\)
\(47\)	1.33586	+	1.81003i	0.194855	+	0.264019i	0.891122	−	0.453763i	\(-0.149919\pi\)
							−0.696267	+	0.717783i	\(0.745157\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.26	✓	672	49.5	odd	42	inner
490.2.w.a.157.26	yes	672	5.2	odd	4	inner
490.2.w.a.353.17	yes	672	1.1	even	1	trivial
490.2.w.a.397.17	yes	672	245.152	even	84	inner