Embedded newform 490.2.w.a.103.13

• Label: 490.2.w.a.103.13
• 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			103.13
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.593820 + 0.804598i) q^{2} +(2.98339 - 0.564488i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(1.30513 + 1.81567i) q^{5} +(-1.31741 + 2.73564i) q^{6} +(-1.78375 + 1.95403i) q^{7} +(0.943883 + 0.330279i) q^{8} +(5.78936 - 2.27216i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	2.98339	−	0.564488i	1.72246	−	0.325907i	0.771737	−	0.635942i	\(-0.219388\pi\)
0.950725	+	0.310034i	\(0.100341\pi\)
\(5\)	1.30513	+	1.81567i	0.583671	+	0.811990i
							\(7\)	−1.78375	+	1.95403i	−0.674196	+	0.738553i
\(11\)	−1.15717	+	2.94842i	−0.348900	+	0.888982i								0.643446	+	0.765491i	\(0.277504\pi\)
							−0.992346	+	0.123490i	\(0.960591\pi\)
\(13\)	1.56429	+	0.176253i	0.433855	+	0.0488837i	0.326191	−	0.945304i	\(-0.394235\pi\)
							0.107663	+	0.994187i	\(0.465663\pi\)
\(17\)	−3.21238	−	0.120199i	−0.779118	−	0.0291525i	−0.355126	−	0.934819i	\(-0.615562\pi\)
							−0.423992	+	0.905666i	\(0.639371\pi\)
\(19\)	0.602629	−	1.04378i	0.138252	−	0.239460i	−0.788583	−	0.614929i	\(-0.789185\pi\)
							0.926835	+	0.375468i	\(0.122518\pi\)
\(23\)	−0.207113	−	5.53522i	−0.0431861	−	1.15417i	−0.837419	−	0.546561i	\(-0.815937\pi\)
							0.794233	−	0.607613i	\(-0.207873\pi\)
\(29\)	−4.85261	+	1.10758i	−0.901107	+	0.205672i	−0.647887	−	0.761737i	\(-0.724347\pi\)
							−0.253220	+	0.967409i	\(0.581490\pi\)
\(31\)	7.18126	−	4.14610i	1.28979	−	0.744662i	0.311175	−	0.950353i	\(-0.399277\pi\)
							0.978617	+	0.205691i	\(0.0659441\pi\)
\(37\)	9.12321	−	4.82176i	1.49985	−	0.792692i	0.502437	−	0.864614i	\(-0.332437\pi\)
							0.997410	+	0.0719215i	\(0.0229131\pi\)
\(41\)	−2.49904	−	5.18930i	−0.390284	−	0.810433i	−0.999843	−	0.0176927i	\(-0.994368\pi\)
							0.609559	−	0.792740i	\(-0.291346\pi\)
\(43\)	−3.70501	−	10.5883i	−0.565009	−	1.61470i	−0.771483	−	0.636250i	\(-0.780485\pi\)
							0.206475	−	0.978452i	\(-0.433801\pi\)
\(47\)	4.33299	+	3.19789i	0.632032	+	0.466461i	0.862117	−	0.506710i	\(-0.169139\pi\)
							−0.230085	+	0.973171i	\(0.573900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.13	✓	672
5.2	odd	4	inner	490.2.w.a.397.2	yes	672
49.10	odd	42	inner	490.2.w.a.353.2	yes	672
245.157	even	84	inner	490.2.w.a.157.13	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.13	✓	672	1.1	even	1	trivial
490.2.w.a.157.13	yes	672	245.157	even	84	inner
490.2.w.a.353.2	yes	672	49.10	odd	42	inner
490.2.w.a.397.2	yes	672	5.2	odd	4	inner