Embedded newform 490.2.w.a.467.19

• Label: 490.2.w.a.467.19
• Level: $490$
• Weight: $2$
• Character: 490.467
• 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			467.19
Character	\(\chi\)	\(=\)	490.467
Dual form			490.2.w.a.213.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.757972 - 0.652287i) q^{2} +(-1.18928 - 0.628551i) q^{3} +(0.149042 - 0.988831i) q^{4} +(0.984568 + 2.00764i) q^{5} +(-1.31143 + 0.299326i) q^{6} +(-2.24227 - 1.40436i) q^{7} +(-0.532032 - 0.846724i) q^{8} +(-0.670657 - 0.983673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(101\)	\(197\)
\(\chi(n)\)	\(e\left(\frac{17}{42}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.757972	−	0.652287i	0.535967	−	0.461237i
							\(3\)	−1.18928	−	0.628551i	−0.686629	−	0.362894i	0.0872859	−	0.996183i	\(-0.472181\pi\)
−0.773915	+	0.633289i	\(0.781704\pi\)
\(5\)	0.984568	+	2.00764i	0.440312	+	0.897845i
							\(7\)	−2.24227	−	1.40436i	−0.847499	−	0.530797i
\(11\)	−3.23735	−	2.20719i	−0.976099	−	0.665493i								−0.0335065	−	0.999438i	\(-0.510667\pi\)
							−0.942592	+	0.333946i	\(0.891620\pi\)
\(13\)	0.511798	+	0.179086i	0.141947	+	0.0496695i	0.400318	−	0.916376i	\(-0.368900\pi\)
							−0.258370	+	0.966046i	\(0.583185\pi\)
\(17\)	−1.92386	−	4.40953i	−0.466604	−	1.06947i	−0.977712	−	0.209951i	\(-0.932670\pi\)
							0.511108	−	0.859516i	\(-0.329235\pi\)
\(19\)	3.28332	−	5.68689i	0.753246	−	1.30466i	−0.192995	−	0.981200i	\(-0.561820\pi\)
							0.946241	−	0.323461i	\(-0.104847\pi\)
\(23\)	−2.67292	−	1.16618i	−0.557342	−	0.243166i	0.102290	−	0.994755i	\(-0.467383\pi\)
							−0.659632	+	0.751589i	\(0.729288\pi\)
\(29\)	4.61475	−	3.68014i	0.856937	−	0.683384i	−0.0930545	−	0.995661i	\(-0.529663\pi\)
							0.949991	+	0.312277i	\(0.101092\pi\)
\(31\)	−6.14532	+	3.54800i	−1.10373	+	0.637240i	−0.937199	−	0.348795i	\(-0.886591\pi\)
							−0.166534	+	0.986036i	\(0.553257\pi\)
\(37\)	0.692799	+	0.511309i	0.113895	+	0.0840586i	0.649279	−	0.760551i	\(-0.275071\pi\)
							−0.535383	+	0.844609i	\(0.679833\pi\)
\(41\)	8.60721	+	1.96454i	1.34422	+	0.306809i	0.833301	−	0.552819i	\(-0.186448\pi\)
							0.510919	+	0.859629i	\(0.329305\pi\)
\(43\)	−5.68970	−	3.57507i	−0.867670	−	0.545193i	0.0230556	−	0.999734i	\(-0.492661\pi\)
							−0.890726	+	0.454541i	\(0.849803\pi\)
\(47\)	1.13698	+	1.32119i	0.165845	+	0.192716i	0.834745	−	0.550637i	\(-0.185615\pi\)
							−0.668899	+	0.743353i	\(0.733234\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.467.19	yes	672
5.3	odd	4	inner	490.2.w.a.173.10	yes	672
49.17	odd	42	inner	490.2.w.a.17.10	✓	672
245.213	even	84	inner	490.2.w.a.213.19	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.17.10	✓	672	49.17	odd	42	inner
490.2.w.a.173.10	yes	672	5.3	odd	4	inner
490.2.w.a.213.19	yes	672	245.213	even	84	inner
490.2.w.a.467.19	yes	672	1.1	even	1	trivial