Embedded newform 490.2.w.a.103.28

• Label: 490.2.w.a.103.28
• 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			103.28
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.593820 - 0.804598i) q^{2} +(3.27887 - 0.620396i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(-1.47394 + 1.68152i) q^{5} +(1.44789 - 3.00658i) q^{6} +(1.49357 + 2.18386i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(7.57348 - 2.97237i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	3.27887	−	0.620396i	1.89306	−	0.358186i	0.896640	−	0.442761i	\(-0.146001\pi\)
0.996417	+	0.0845752i	\(0.0269533\pi\)
\(5\)	−1.47394	+	1.68152i	−0.659167	+	0.751997i
							\(7\)	1.49357	+	2.18386i	0.564515	+	0.825423i
\(11\)	0.705140	−	1.79667i	0.212608	−	0.541716i								−0.784292	−	0.620391i	\(-0.786974\pi\)
							0.996900	+	0.0786754i	\(0.0250691\pi\)
\(13\)	−1.51007	−	0.170144i	−0.418817	−	0.0471894i	−0.0999585	−	0.994992i	\(-0.531871\pi\)
							−0.318859	+	0.947802i	\(0.603300\pi\)
\(17\)	−5.67076	−	0.212185i	−1.37536	−	0.0514624i	−0.660552	−	0.750781i	\(-0.729678\pi\)
							−0.714811	+	0.699318i	\(0.753487\pi\)
\(19\)	−0.646394	+	1.11959i	−0.148293	+	0.256851i	−0.930597	−	0.366046i	\(-0.880711\pi\)
							0.782304	+	0.622897i	\(0.214045\pi\)
\(23\)	−0.0222878	−	0.595655i	−0.00464734	−	0.124203i	−0.999693	−	0.0247824i	\(-0.992111\pi\)
							0.995046	−	0.0994203i	\(-0.0316988\pi\)
\(29\)	6.01453	−	1.37278i	1.11687	−	0.254918i	0.376035	−	0.926605i	\(-0.377287\pi\)
							0.740835	+	0.671687i	\(0.234430\pi\)
\(31\)	−6.00054	+	3.46441i	−1.07773	+	0.622227i	−0.930283	−	0.366843i	\(-0.880439\pi\)
							−0.147446	+	0.989070i	\(0.547105\pi\)
\(37\)	−4.67641	+	2.47156i	−0.768797	+	0.406321i	−0.805190	−	0.593016i	\(-0.797937\pi\)
							0.0363932	+	0.999338i	\(0.488413\pi\)
\(41\)	−3.61663	−	7.51000i	−0.564822	−	1.17287i	−0.966401	−	0.257041i	\(-0.917252\pi\)
							0.401578	−	0.915825i	\(-0.368462\pi\)
\(43\)	2.85475	+	8.15842i	0.435346	+	1.24415i	0.927561	+	0.373673i	\(0.121902\pi\)
							−0.492215	+	0.870474i	\(0.663813\pi\)
\(47\)	−10.0406	−	7.41033i	−1.46458	−	1.08091i	−0.978322	−	0.207090i	\(-0.933601\pi\)
							−0.486255	−	0.873817i	\(-0.661638\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.28	✓	672
5.2	odd	4	inner	490.2.w.a.397.15	yes	672
49.10	odd	42	inner	490.2.w.a.353.15	yes	672
245.157	even	84	inner	490.2.w.a.157.28	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.28	✓	672	1.1	even	1	trivial
490.2.w.a.157.28	yes	672	245.157	even	84	inner
490.2.w.a.353.15	yes	672	49.10	odd	42	inner
490.2.w.a.397.15	yes	672	5.2	odd	4	inner