Embedded newform 731.2.p.a.594.19

• Label: 731.2.p.a.594.19
• 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			594.19
Character	\(\chi\)	\(=\)	731.594
Dual form			731.2.p.a.16.19

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

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

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	−0.810922	−	0.585154i	\(-0.801034\pi\)
							−0.476727	+	0.879051i	\(0.658177\pi\)
\(5\)	−3.41321	−	2.72194i	−1.52643	−	1.21729i	−0.898460	−	0.439055i	\(-0.855313\pi\)
							−0.627973	−	0.778235i	\(-0.716115\pi\)
\(7\)			3.16066i			1.19462i	0.802012	+	0.597308i	\(0.203763\pi\)
							−0.802012	+	0.597308i	\(0.796237\pi\)
\(11\)	−1.29978	−	2.69901i	−0.391897	−	0.813782i	−0.999804	−	0.0197746i	\(-0.993705\pi\)
							0.607908	−	0.794008i	\(-0.292009\pi\)
\(13\)	1.50839	−	1.89146i	0.418351	−	0.524596i	−0.527343	−	0.849652i	\(-0.676812\pi\)
							0.945695	+	0.325056i	\(0.105383\pi\)
\(17\)	−1.17089	−	3.95336i	−0.283982	−	0.958829i
							\(19\)	−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	\(0.0653312\pi\)
							−0.451062	+	0.892493i	\(0.648955\pi\)
\(29\)	−3.59622	−	0.820814i	−0.667802	−	0.152421i	−0.124837	−	0.992177i	\(-0.539841\pi\)
							−0.542964	+	0.839756i	\(0.682698\pi\)
\(31\)	7.49087	+	1.70974i	1.34540	+	0.307079i	0.833763	−	0.552123i	\(-0.186182\pi\)
							0.511637	+	0.859202i	\(0.329039\pi\)
\(37\)			10.7006i			1.75916i	0.475747	+	0.879582i	\(0.342178\pi\)
							−0.475747	+	0.879582i	\(0.657822\pi\)
\(41\)	−1.41945	−	0.323980i	−0.221681	−	0.0505971i	0.110238	−	0.993905i	\(-0.464839\pi\)
							−0.331918	+	0.943308i	\(0.607696\pi\)
\(43\)	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.594.19	yes	384
17.16	even	2	inner	731.2.p.a.594.20	yes	384
43.16	even	7	inner	731.2.p.a.16.20	yes	384
731.16	even	14	inner	731.2.p.a.16.19	✓	384

    

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