Embedded newform 731.2.p.a.16.9

• Label: 731.2.p.a.16.9
• 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			16.9
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.501288 - 2.19629i) q^{2} +(-1.02764 - 0.234553i) q^{3} +(-2.77045 + 1.33418i) q^{4} +(1.62476 - 1.29571i) q^{5} +2.37458i q^{6} +4.11644i q^{7} +(1.50987 + 1.89332i) q^{8} +(-1.70187 - 0.819578i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.501288	−	2.19629i	−0.354464	−	1.55301i	−0.766745	−	0.641952i	\(-0.778125\pi\)
							0.412280	−	0.911057i	\(-0.364732\pi\)
\(3\)	−1.02764	−	0.234553i	−0.593310	−	0.135419i	−0.0846858	−	0.996408i	\(-0.526989\pi\)
							−0.508624	+	0.860989i	\(0.669846\pi\)
\(5\)	1.62476	−	1.29571i	0.726617	−	0.579457i	−0.188776	−	0.982020i	\(-0.560452\pi\)
							0.915392	+	0.402563i	\(0.131881\pi\)
\(7\)			4.11644i			1.55587i	0.628345	+	0.777934i	\(0.283732\pi\)
							−0.628345	+	0.777934i	\(0.716268\pi\)
\(11\)	−0.236630	+	0.491367i	−0.0713466	+	0.148153i	−0.933592	−	0.358338i	\(-0.883343\pi\)
							0.862245	+	0.506491i	\(0.169058\pi\)
\(13\)	2.43613	+	3.05481i	0.675662	+	0.847253i	0.994947	−	0.100406i	\(-0.0320141\pi\)
							−0.319285	+	0.947659i	\(0.603443\pi\)
\(17\)	3.68399	+	1.85155i	0.893498	+	0.449068i
							\(19\)	−1.60375	+	0.772326i	−0.367926	+	0.177184i	−0.608706	−	0.793396i	\(-0.708311\pi\)
0.240780	+	0.970580i	\(0.422597\pi\)
\(23\)	−0.956800	+	1.98682i	−0.199507	+	0.414280i	−0.976587	−	0.215121i	\(-0.930985\pi\)
							0.777081	+	0.629401i	\(0.216700\pi\)
\(29\)	3.76745	−	0.859896i	0.699598	−	0.159679i	0.142094	−	0.989853i	\(-0.454616\pi\)
							0.557504	+	0.830174i	\(0.311759\pi\)
\(31\)	6.82752	−	1.55834i	1.22626	−	0.279886i	0.440122	−	0.897938i	\(-0.354935\pi\)
							0.786137	+	0.618052i	\(0.212078\pi\)
\(37\)		−	0.516885i		−	0.0849754i	−0.999097	−	0.0424877i	\(-0.986472\pi\)
							0.999097	−	0.0424877i	\(-0.0135283\pi\)
\(41\)	4.31959	−	0.985917i	0.674606	−	0.153974i	0.128524	−	0.991706i	\(-0.458976\pi\)
							0.546082	+	0.837732i	\(0.316119\pi\)
\(43\)	−5.74959	+	3.15313i	−0.876804	+	0.480848i
							\(47\)	−7.94766	+	3.82739i	−1.15929	+	0.558283i	−0.911810	−	0.410611i	\(-0.865315\pi\)
−0.247476	+	0.968894i	\(0.579601\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.16.9	✓	384
17.16	even	2	inner	731.2.p.a.16.10	yes	384
43.35	even	7	inner	731.2.p.a.594.10	yes	384
731.594	even	14	inner	731.2.p.a.594.9	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.9	✓	384	1.1	even	1	trivial
731.2.p.a.16.10	yes	384	17.16	even	2	inner
731.2.p.a.594.9	yes	384	731.594	even	14	inner
731.2.p.a.594.10	yes	384	43.35	even	7	inner