Embedded newform 731.2.p.a.16.3

• Label: 731.2.p.a.16.3
• 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			16.3
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.594326 - 2.60391i) q^{2} +(-2.90090 - 0.662112i) q^{3} +(-4.62520 + 2.22738i) q^{4} +(-1.09262 + 0.871338i) q^{5} +7.94720i q^{6} -2.92823i q^{7} +(5.21824 + 6.54347i) q^{8} +(5.27392 + 2.53979i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.594326	−	2.60391i	−0.420252	−	1.84124i	−0.530968	−	0.847392i	\(-0.678172\pi\)
							0.110716	−	0.993852i	\(-0.464686\pi\)
\(3\)	−2.90090	−	0.662112i	−1.67484	−	0.382270i	−0.723475	−	0.690350i	\(-0.757456\pi\)
							−0.951360	+	0.308080i	\(0.900313\pi\)
\(5\)	−1.09262	+	0.871338i	−0.488636	+	0.389674i	−0.836583	−	0.547840i	\(-0.815450\pi\)
							0.347947	+	0.937514i	\(0.386879\pi\)
\(7\)		−	2.92823i		−	1.10677i	−0.832927	−	0.553383i	\(-0.813337\pi\)
							0.832927	−	0.553383i	\(-0.186663\pi\)
\(11\)	2.10177	−	4.36437i	0.633708	−	1.31591i	−0.298647	−	0.954364i	\(-0.596535\pi\)
							0.932355	−	0.361544i	\(-0.117750\pi\)
\(13\)	3.19001	+	4.00014i	0.884749	+	1.10944i	0.993325	+	0.115352i	\(0.0367996\pi\)
							−0.108576	+	0.994088i	\(0.534629\pi\)
\(17\)	4.12246	+	0.0729413i	0.999844	+	0.0176909i
							\(19\)	0.341983	−	0.164690i	0.0784562	−	0.0377825i	−0.394244	−	0.919006i	\(-0.628994\pi\)
0.472700	+	0.881223i	\(0.343279\pi\)
\(23\)	−2.67410	+	5.55284i	−0.557589	+	1.15785i	0.411562	+	0.911382i	\(0.364983\pi\)
							−0.969152	+	0.246465i	\(0.920731\pi\)
\(29\)	−1.64734	+	0.375994i	−0.305903	+	0.0698203i	−0.372717	−	0.927945i	\(-0.621574\pi\)
							0.0668141	+	0.997765i	\(0.478717\pi\)
\(31\)	6.07677	−	1.38698i	1.09142	−	0.249110i	0.361315	−	0.932444i	\(-0.382328\pi\)
							0.730106	+	0.683334i	\(0.239471\pi\)
\(37\)			2.44803i			0.402454i	0.979545	+	0.201227i	\(0.0644929\pi\)
							−0.979545	+	0.201227i	\(0.935507\pi\)
\(41\)	−0.131657	+	0.0300498i	−0.0205614	+	0.00469300i	−0.232789	−	0.972527i	\(-0.574785\pi\)
							0.212228	+	0.977220i	\(0.431928\pi\)
\(43\)	2.45227	−	6.08164i	0.373967	−	0.927442i
							\(47\)	−7.04152	+	3.39102i	−1.02711	+	0.494631i	−0.870053	−	0.492958i	\(-0.835915\pi\)
−0.157059	+	0.987589i	\(0.550201\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.16.3	✓	384
17.16	even	2	inner	731.2.p.a.16.4	yes	384
43.35	even	7	inner	731.2.p.a.594.4	yes	384
731.594	even	14	inner	731.2.p.a.594.3	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.3	✓	384	1.1	even	1	trivial
731.2.p.a.16.4	yes	384	17.16	even	2	inner
731.2.p.a.594.3	yes	384	731.594	even	14	inner
731.2.p.a.594.4	yes	384	43.35	even	7	inner