Embedded newform 731.2.p.a.16.10

• Label: 731.2.p.a.16.10
• 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			16.10
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.10

$q$-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	0.508624	−	0.860989i	\(-0.330154\pi\)
							0.0846858	+	0.996408i	\(0.473011\pi\)
\(5\)	−1.62476	+	1.29571i	−0.726617	+	0.579457i	−0.915392	−	0.402563i	\(-0.868119\pi\)
							0.188776	+	0.982020i	\(0.439548\pi\)
\(7\)		−	4.11644i		−	1.55587i	−0.628345	−	0.777934i	\(-0.716268\pi\)
							0.628345	−	0.777934i	\(-0.283732\pi\)
\(11\)	0.236630	−	0.491367i	0.0713466	−	0.148153i	−0.862245	−	0.506491i	\(-0.830942\pi\)
							0.933592	+	0.358338i	\(0.116657\pi\)
\(13\)	2.43613	+	3.05481i	0.675662	+	0.847253i	0.994947	−	0.100406i	\(-0.0320141\pi\)
							−0.319285	+	0.947659i	\(0.603443\pi\)
\(17\)	−2.62490	−	3.17961i	−0.636631	−	0.771169i
							\(19\)	−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	−0.777081	−	0.629401i	\(-0.783300\pi\)
							0.976587	+	0.215121i	\(0.0690147\pi\)
\(29\)	−3.76745	+	0.859896i	−0.699598	+	0.159679i	−0.557504	−	0.830174i	\(-0.688241\pi\)
							−0.142094	+	0.989853i	\(0.545384\pi\)
\(31\)	−6.82752	+	1.55834i	−1.22626	+	0.279886i	−0.786137	−	0.618052i	\(-0.787922\pi\)
							−0.440122	+	0.897938i	\(0.645065\pi\)
\(37\)			0.516885i			0.0849754i	0.999097	+	0.0424877i	\(0.0135283\pi\)
							−0.999097	+	0.0424877i	\(0.986472\pi\)
\(41\)	−4.31959	+	0.985917i	−0.674606	+	0.153974i	−0.546082	−	0.837732i	\(-0.683881\pi\)
							−0.128524	+	0.991706i	\(0.541024\pi\)
\(43\)	−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.16.10	yes	384
17.16	even	2	inner	731.2.p.a.16.9	✓	384
43.35	even	7	inner	731.2.p.a.594.9	yes	384
731.594	even	14	inner	731.2.p.a.594.10	yes	384

    

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