Embedded newform 490.2.w.a.103.4

• Label: 490.2.w.a.103.4
• 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			103.4
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.593820 + 0.804598i) q^{2} +(-1.47482 + 0.279051i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(1.17673 - 1.90140i) q^{5} +(0.651255 - 1.35234i) q^{6} +(0.808220 - 2.51928i) q^{7} +(0.943883 + 0.330279i) q^{8} +(-0.695393 + 0.272922i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	−1.47482	+	0.279051i	−0.851488	+	0.161110i	−0.593297	−	0.804984i	\(-0.702174\pi\)
−0.258191	+	0.966094i	\(0.583126\pi\)
\(5\)	1.17673	−	1.90140i	0.526248	−	0.850331i
							\(7\)	0.808220	−	2.51928i	0.305479	−	0.952199i
\(11\)	−0.542772	+	1.38296i	−0.163652	+	0.416978i								−0.989052	−	0.147571i	\(-0.952855\pi\)
							0.825400	+	0.564549i	\(0.190950\pi\)
\(13\)	−3.21013	−	0.361695i	−0.890331	−	0.100316i	−0.345065	−	0.938579i	\(-0.612143\pi\)
							−0.545267	+	0.838263i	\(0.683572\pi\)
\(17\)	−3.97900	−	0.148884i	−0.965049	−	0.0361096i	−0.449781	−	0.893139i	\(-0.648498\pi\)
							−0.515268	+	0.857029i	\(0.672308\pi\)
\(19\)	−3.31077	+	5.73443i	−0.759543	+	1.31557i	0.183540	+	0.983012i	\(0.441244\pi\)
							−0.943084	+	0.332555i	\(0.892089\pi\)
\(23\)	−0.164805	−	4.40451i	−0.0343642	−	0.918404i	−0.904727	−	0.425992i	\(-0.859925\pi\)
							0.870363	−	0.492411i	\(-0.163884\pi\)
\(29\)	−1.45233	+	0.331485i	−0.269691	+	0.0615553i	−0.355228	−	0.934780i	\(-0.615597\pi\)
							0.0855366	+	0.996335i	\(0.472740\pi\)
\(31\)	−7.94116	+	4.58483i	−1.42627	+	0.823460i	−0.996824	−	0.0796306i	\(-0.974626\pi\)
							−0.429450	+	0.903091i	\(0.641293\pi\)
\(37\)	6.97165	−	3.68462i	1.14613	−	0.605748i	0.217247	−	0.976117i	\(-0.430292\pi\)
							0.928885	+	0.370368i	\(0.120768\pi\)
\(41\)	−3.11333	−	6.46489i	−0.486220	−	1.00965i	−0.989366	−	0.145445i	\(-0.953539\pi\)
							0.503146	−	0.864201i	\(-0.332176\pi\)
\(43\)	1.81178	+	5.17776i	0.276293	+	0.789601i	0.995496	+	0.0948010i	\(0.0302215\pi\)
							−0.719203	+	0.694800i	\(0.755493\pi\)
\(47\)	−2.94656	−	2.17466i	−0.429800	−	0.317207i	0.357347	−	0.933972i	\(-0.383681\pi\)
							−0.787148	+	0.616764i	\(0.788443\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.4	✓	672
5.2	odd	4	inner	490.2.w.a.397.11	yes	672
49.10	odd	42	inner	490.2.w.a.353.11	yes	672
245.157	even	84	inner	490.2.w.a.157.4	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.4	✓	672	1.1	even	1	trivial
490.2.w.a.157.4	yes	672	245.157	even	84	inner
490.2.w.a.353.11	yes	672	49.10	odd	42	inner
490.2.w.a.397.11	yes	672	5.2	odd	4	inner