Embedded newform 490.2.w.a.103.23

• Label: 490.2.w.a.103.23
• 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			103.23
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.593820 - 0.804598i) q^{2} +(0.000603296 - 0.000114150i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(-1.17086 + 1.90501i) q^{5} +(0.000266405 - 0.000553195i) q^{6} +(-2.50652 + 0.846959i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(-2.79262 + 1.09602i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	0.000603296	0	0.000114150i	0.000348313	0	6.59044e-5i	−0.185737	−	0.982599i	\(-0.559467\pi\)
0.186086	+	0.982534i	\(0.440420\pi\)
\(5\)	−1.17086	+	1.90501i	−0.523626	+	0.851948i
							\(7\)	−2.50652	+	0.846959i	−0.947377	+	0.320120i
\(11\)	0.156322	−	0.398301i	0.0471327	−	0.120092i								−0.905369	−	0.424626i	\(-0.860406\pi\)
							0.952501	+	0.304534i	\(0.0985007\pi\)
\(13\)	−0.598790	−	0.0674675i	−0.166075	−	0.0187121i	0.0285356	−	0.999593i	\(-0.490916\pi\)
							−0.194610	+	0.980881i	\(0.562344\pi\)
\(17\)	−1.75818	−	0.0657864i	−0.426421	−	0.0159555i	−0.176680	−	0.984268i	\(-0.556536\pi\)
							−0.249741	+	0.968313i	\(0.580345\pi\)
\(19\)	−3.69043	+	6.39202i	−0.846643	+	1.46643i	0.0375434	+	0.999295i	\(0.488047\pi\)
							−0.884187	+	0.467134i	\(0.845287\pi\)
\(23\)	0.0107398	+	0.287027i	0.00223940	+	0.0598492i	0.999972	−	0.00744982i	\(-0.00237137\pi\)
							−0.997733	+	0.0672990i	\(0.978562\pi\)
\(29\)	−1.89914	+	0.433466i	−0.352661	+	0.0804926i	−0.395182	−	0.918603i	\(-0.629318\pi\)
							0.0425205	+	0.999096i	\(0.486461\pi\)
\(31\)	1.61357	−	0.931593i	0.289805	−	0.167319i	−0.348049	−	0.937476i	\(-0.613156\pi\)
							0.637854	+	0.770157i	\(0.279822\pi\)
\(37\)	5.88844	−	3.11213i	0.968054	−	0.511632i	0.0930051	−	0.995666i	\(-0.470353\pi\)
							0.875049	+	0.484034i	\(0.160829\pi\)
\(41\)	1.46758	+	3.04746i	0.229197	+	0.475933i	0.983573	−	0.180510i	\(-0.0577747\pi\)
							−0.754376	+	0.656443i	\(0.772060\pi\)
\(43\)	0.173671	+	0.496324i	0.0264846	+	0.0756887i	0.956367	−	0.292169i	\(-0.0943770\pi\)
							−0.929882	+	0.367858i	\(0.880091\pi\)
\(47\)	4.38717	+	3.23788i	0.639934	+	0.472293i	0.864813	−	0.502094i	\(-0.167437\pi\)
							−0.224879	+	0.974387i	\(0.572199\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.23	✓	672
5.2	odd	4	inner	490.2.w.a.397.20	yes	672
49.10	odd	42	inner	490.2.w.a.353.20	yes	672
245.157	even	84	inner	490.2.w.a.157.23	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.23	✓	672	1.1	even	1	trivial
490.2.w.a.157.23	yes	672	245.157	even	84	inner
490.2.w.a.353.20	yes	672	49.10	odd	42	inner
490.2.w.a.397.20	yes	672	5.2	odd	4	inner