Embedded newform 731.2.p.a.16.12

• Label: 731.2.p.a.16.12
• 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			16.12
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.473651 - 2.07520i) q^{2} +(3.20126 + 0.730666i) q^{3} +(-2.28018 + 1.09808i) q^{4} +(0.606394 - 0.483583i) q^{5} -6.98933i q^{6} -4.01427i q^{7} +(0.704462 + 0.883367i) q^{8} +(7.01126 + 3.37644i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.473651	−	2.07520i	−0.334922	−	1.46739i	−0.809471	−	0.587160i	\(-0.800246\pi\)
							0.474548	−	0.880229i	\(-0.342611\pi\)
\(3\)	3.20126	+	0.730666i	1.84825	+	0.421850i	0.995019	−	0.0996898i	\(-0.0317850\pi\)
							0.853227	+	0.521540i	\(0.174642\pi\)
\(5\)	0.606394	−	0.483583i	0.271188	−	0.216265i	−0.478448	−	0.878116i	\(-0.658800\pi\)
							0.749635	+	0.661851i	\(0.230229\pi\)
\(7\)		−	4.01427i		−	1.51725i	−0.651528	−	0.758625i	\(-0.725872\pi\)
							0.651528	−	0.758625i	\(-0.274128\pi\)
\(11\)	0.0655147	−	0.136043i	0.0197534	−	0.0410184i	−0.890859	−	0.454281i	\(-0.849896\pi\)
							0.910612	+	0.413262i	\(0.135611\pi\)
\(13\)	−2.82091	−	3.53732i	−0.782381	−	0.981075i	−0.999987	−	0.00503814i	\(-0.998396\pi\)
							0.217606	−	0.976037i	\(-0.430175\pi\)
\(17\)	−1.81683	+	3.70123i	−0.440646	+	0.897681i
							\(19\)	5.48249	−	2.64023i	1.25777	−	0.605710i	0.318186	−	0.948028i	\(-0.396926\pi\)
0.939584	+	0.342318i	\(0.111212\pi\)
\(23\)	−2.64022	+	5.48246i	−0.550523	+	1.14317i	0.421182	+	0.906976i	\(0.361615\pi\)
							−0.971705	+	0.236197i	\(0.924099\pi\)
\(29\)	3.86077	−	0.881195i	0.716927	−	0.163634i	0.151530	−	0.988453i	\(-0.451580\pi\)
							0.565397	+	0.824819i	\(0.308723\pi\)
\(31\)	−5.46981	+	1.24845i	−0.982406	+	0.224228i	−0.683417	−	0.730029i	\(-0.739507\pi\)
							−0.298989	+	0.954256i	\(0.596650\pi\)
\(37\)		−	1.77438i		−	0.291706i	−0.989306	−	0.145853i	\(-0.953407\pi\)
							0.989306	−	0.145853i	\(-0.0465926\pi\)
\(41\)	−0.498334	+	0.113742i	−0.0778268	+	0.0177635i	−0.261257	−	0.965269i	\(-0.584137\pi\)
							0.183430	+	0.983033i	\(0.441280\pi\)
\(43\)	−5.20221	+	3.99212i	−0.793329	+	0.608793i
							\(47\)	6.46678	−	3.11424i	0.943277	−	0.454258i	0.101953	−	0.994789i	\(-0.467491\pi\)
0.841324	+	0.540531i	\(0.181777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.16.12	yes	384
17.16	even	2	inner	731.2.p.a.16.11	✓	384
43.35	even	7	inner	731.2.p.a.594.11	yes	384
731.594	even	14	inner	731.2.p.a.594.12	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.11	✓	384	17.16	even	2	inner
731.2.p.a.16.12	yes	384	1.1	even	1	trivial
731.2.p.a.594.11	yes	384	43.35	even	7	inner
731.2.p.a.594.12	yes	384	731.594	even	14	inner