Embedded newform 731.2.p.a.16.19

• Label: 731.2.p.a.16.19
• Level: $731$
• Weight: $2$
• Character: 731.16
• 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			16.19
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.19

$q$-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 \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)

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	−0.945231	−	0.326403i	\(-0.894163\pi\)
							0.710003	−	0.704199i	\(-0.248694\pi\)
\(3\)	−2.23027	−	0.509045i	−1.28765	−	0.293898i	−0.476727	−	0.879051i	\(-0.658177\pi\)
							−0.810922	+	0.585154i	\(0.801034\pi\)
\(5\)	−3.41321	+	2.72194i	−1.52643	+	1.21729i	−0.627973	+	0.778235i	\(0.716115\pi\)
							−0.898460	+	0.439055i	\(0.855313\pi\)
\(7\)		−	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	0.607908	+	0.794008i	\(0.292009\pi\)
							−0.999804	+	0.0197746i	\(0.993705\pi\)
\(13\)	1.50839	+	1.89146i	0.418351	+	0.524596i	0.945695	−	0.325056i	\(-0.105383\pi\)
							−0.527343	+	0.849652i	\(0.676812\pi\)
\(17\)	−1.17089	+	3.95336i	−0.283982	+	0.958829i
							\(19\)	−3.69742	+	1.78059i	−0.848247	+	0.408494i	−0.806927	−	0.590652i	\(-0.798871\pi\)
−0.0413206	+	0.999146i	\(0.513157\pi\)
\(23\)	2.53196	−	5.25766i	0.527950	−	1.09630i	−0.451062	−	0.892493i	\(-0.648955\pi\)
							0.979011	−	0.203806i	\(-0.0653312\pi\)
\(29\)	−3.59622	+	0.820814i	−0.667802	+	0.152421i	−0.542964	−	0.839756i	\(-0.682698\pi\)
							−0.124837	+	0.992177i	\(0.539841\pi\)
\(31\)	7.49087	−	1.70974i	1.34540	−	0.307079i	0.511637	−	0.859202i	\(-0.329039\pi\)
							0.833763	+	0.552123i	\(0.186182\pi\)
\(37\)		−	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	\(-0.607696\pi\)
							0.110238	+	0.993905i	\(0.464839\pi\)
\(43\)	2.40268	+	6.10140i	0.366405	+	0.930455i
							\(47\)	9.11023	−	4.38726i	1.32886	−	0.639947i	0.371393	−	0.928476i	\(-0.378880\pi\)
0.957471	+	0.288529i	\(0.0931661\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.19	✓	384	1.1	even	1	trivial
731.2.p.a.16.20	yes	384	17.16	even	2	inner
731.2.p.a.594.19	yes	384	731.594	even	14	inner
731.2.p.a.594.20	yes	384	43.35	even	7	inner