Embedded newform 490.2.w.a.103.24

• Label: 490.2.w.a.103.24
• 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			103.24
Character	\(\chi\)	\(=\)	490.103
Dual form			490.2.w.a.157.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.593820 - 0.804598i) q^{2} +(1.65254 - 0.312678i) q^{3} +(-0.294755 - 0.955573i) q^{4} +(2.14046 + 0.646873i) q^{5} +(0.729733 - 1.51531i) q^{6} +(-1.08917 + 2.41116i) q^{7} +(-0.943883 - 0.330279i) q^{8} +(-0.159496 + 0.0625976i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 672 q + 12 q^{5} - 28 q^{6} - 8 q^{7}+O(q^{10}) \)

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
							\(3\)	1.65254	−	0.312678i	0.954095	−	0.180525i	0.314518	−	0.949252i	\(-0.398157\pi\)
0.639577	+	0.768727i	\(0.279109\pi\)
\(5\)	2.14046	+	0.646873i	0.957241	+	0.289290i
							\(7\)	−1.08917	+	2.41116i	−0.411669	+	0.911334i
\(11\)	1.58469	−	4.03772i	0.477802	−	1.21742i								−0.465371	−	0.885116i	\(-0.654079\pi\)
							0.943173	−	0.332303i	\(-0.107826\pi\)
\(13\)	2.32661	+	0.262146i	0.645286	+	0.0727063i	0.428542	−	0.903522i	\(-0.359028\pi\)
							0.216745	+	0.976228i	\(0.430456\pi\)
\(17\)	0.779339	+	0.0291608i	0.189017	+	0.00707253i	0.131732	−	0.991285i	\(-0.457946\pi\)
							0.0572851	+	0.998358i	\(0.481756\pi\)
\(19\)	3.47180	−	6.01333i	0.796485	−	1.37955i	−0.125406	−	0.992105i	\(-0.540023\pi\)
							0.921892	−	0.387448i	\(-0.126643\pi\)
\(23\)	0.123683	+	3.30550i	0.0257897	+	0.689243i	0.950892	+	0.309524i	\(0.100170\pi\)
							−0.925102	+	0.379719i	\(0.876021\pi\)
\(29\)	−8.83692	+	2.01697i	−1.64097	+	0.374542i	−0.940675	−	0.339308i	\(-0.889807\pi\)
							−0.700299	+	0.713850i	\(0.746950\pi\)
\(31\)	−2.15200	+	1.24246i	−0.386511	+	0.223152i	−0.680647	−	0.732611i	\(-0.738301\pi\)
							0.294137	+	0.955763i	\(0.404968\pi\)
\(37\)	−4.10585	+	2.17001i	−0.674998	+	0.356747i	−0.769382	−	0.638789i	\(-0.779436\pi\)
							0.0943839	+	0.995536i	\(0.469912\pi\)
\(41\)	0.857819	+	1.78128i	0.133969	+	0.278189i	0.957152	−	0.289587i	\(-0.0935181\pi\)
							−0.823183	+	0.567777i	\(0.807804\pi\)
\(43\)	−0.471365	−	1.34709i	−0.0718826	−	0.205429i	0.902337	−	0.431032i	\(-0.141850\pi\)
							−0.974219	+	0.225603i	\(0.927565\pi\)
\(47\)	−0.730414	−	0.539070i	−0.106542	−	0.0786315i	0.539246	−	0.842148i	\(-0.318709\pi\)
							−0.645788	+	0.763517i	\(0.723471\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	490.2.w.a.103.24	✓	672
5.2	odd	4	inner	490.2.w.a.397.19	yes	672
49.10	odd	42	inner	490.2.w.a.353.19	yes	672
245.157	even	84	inner	490.2.w.a.157.24	yes	672

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.w.a.103.24	✓	672	1.1	even	1	trivial
490.2.w.a.157.24	yes	672	245.157	even	84	inner
490.2.w.a.353.19	yes	672	49.10	odd	42	inner
490.2.w.a.397.19	yes	672	5.2	odd	4	inner