Embedded newform 731.2.p.a.594.20

• Label: 731.2.p.a.594.20
• Level: $731$
• Weight: $2$
• Character: 731.594
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(64\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			594.20
Character	\(\chi\)	\(=\)	731.594
Dual form			731.2.p.a.16.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.332663 + 1.45749i) q^{2} +(2.23027 - 0.509045i) q^{3} +(-0.211677 - 0.101938i) q^{4} +(3.41321 + 2.72194i) q^{5} +3.41994i q^{6} -3.16066i q^{7} +(-1.64521 + 2.06303i) q^{8} +(2.01209 - 0.968970i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.332663	+	1.45749i	−0.235228	+	1.03060i	0.710003	+	0.704199i	\(0.248694\pi\)
							−0.945231	+	0.326403i	\(0.894163\pi\)
\(3\)	2.23027	−	0.509045i	1.28765	−	0.293898i	0.476727	−	0.879051i	\(-0.341823\pi\)
							0.810922	+	0.585154i	\(0.198966\pi\)
\(5\)	3.41321	+	2.72194i	1.52643	+	1.21729i	0.898460	+	0.439055i	\(0.144687\pi\)
							0.627973	+	0.778235i	\(0.283885\pi\)
\(7\)		−	3.16066i		−	1.19462i	−0.802012	−	0.597308i	\(-0.796237\pi\)
							0.802012	−	0.597308i	\(-0.203763\pi\)
\(11\)	1.29978	+	2.69901i	0.391897	+	0.813782i	0.999804	+	0.0197746i	\(0.00629485\pi\)
							−0.607908	+	0.794008i	\(0.707991\pi\)
\(13\)	1.50839	−	1.89146i	0.418351	−	0.524596i	−0.527343	−	0.849652i	\(-0.676812\pi\)
							0.945695	+	0.325056i	\(0.105383\pi\)
\(17\)	−3.59369	−	2.02124i	−0.871598	−	0.490222i
							\(19\)	−3.69742	−	1.78059i	−0.848247	−	0.408494i	−0.0413206	−	0.999146i	\(-0.513157\pi\)
−0.806927	+	0.590652i	\(0.798871\pi\)
\(23\)	−2.53196	−	5.25766i	−0.527950	−	1.09630i	−0.979011	−	0.203806i	\(-0.934669\pi\)
							0.451062	−	0.892493i	\(-0.351045\pi\)
\(29\)	3.59622	+	0.820814i	0.667802	+	0.152421i	0.542964	−	0.839756i	\(-0.317302\pi\)
							0.124837	+	0.992177i	\(0.460159\pi\)
\(31\)	−7.49087	−	1.70974i	−1.34540	−	0.307079i	−0.511637	−	0.859202i	\(-0.670961\pi\)
							−0.833763	+	0.552123i	\(0.813818\pi\)
\(37\)		−	10.7006i		−	1.75916i	−0.475747	−	0.879582i	\(-0.657822\pi\)
							0.475747	−	0.879582i	\(-0.342178\pi\)
\(41\)	1.41945	+	0.323980i	0.221681	+	0.0505971i	0.331918	−	0.943308i	\(-0.392304\pi\)
							−0.110238	+	0.993905i	\(0.535161\pi\)
\(43\)	2.40268	−	6.10140i	0.366405	−	0.930455i
							\(47\)	9.11023	+	4.38726i	1.32886	+	0.639947i	0.957471	−	0.288529i	\(-0.0931661\pi\)
0.371393	+	0.928476i	\(0.378880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.594.20	yes	384
17.16	even	2	inner	731.2.p.a.594.19	yes	384
43.16	even	7	inner	731.2.p.a.16.19	✓	384
731.16	even	14	inner	731.2.p.a.16.20	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.19	✓	384	43.16	even	7	inner
731.2.p.a.16.20	yes	384	731.16	even	14	inner
731.2.p.a.594.19	yes	384	17.16	even	2	inner
731.2.p.a.594.20	yes	384	1.1	even	1	trivial