Embedded newform 731.2.s.b.214.3

• Label: 731.2.s.b.214.3
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			214.3
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.43313 - 1.00784i) q^{2} +(-0.227777 + 0.340892i) q^{3} +(3.49018 + 3.49018i) q^{4} +(-0.210988 - 1.06071i) q^{5} +(0.897773 - 0.599873i) q^{6} +(-0.282349 + 1.41946i) q^{7} +(-2.95886 - 7.14332i) q^{8} +(1.08373 + 2.61634i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.43313	−	1.00784i	−1.72048	−	0.712647i	−0.999812	−	0.0193699i	\(-0.993834\pi\)
							−0.720671	−	0.693278i	\(-0.756166\pi\)
\(3\)	−0.227777	+	0.340892i	−0.131507	+	0.196814i	−0.891379	−	0.453258i	\(-0.850261\pi\)
							0.759872	+	0.650072i	\(0.225261\pi\)
\(5\)	−0.210988	−	1.06071i	−0.0943568	−	0.474364i	−0.998853	−	0.0478820i	\(-0.984753\pi\)
							0.904496	−	0.426482i	\(-0.140247\pi\)
\(7\)	−0.282349	+	1.41946i	−0.106718	+	0.536506i	0.890029	+	0.455905i	\(0.150684\pi\)
							−0.996746	+	0.0806016i	\(0.974316\pi\)
\(11\)	−2.12378	−	3.17846i	−0.640343	−	0.958341i	−0.999684	−	0.0251347i	\(-0.991999\pi\)
							0.359341	−	0.933206i	\(-0.383001\pi\)
\(13\)	−0.324587	+	0.324587i	−0.0900243	+	0.0900243i	−0.750685	−	0.660661i	\(-0.770276\pi\)
							0.660661	+	0.750685i	\(0.270276\pi\)
\(17\)	−0.221689	−	4.11714i	−0.0537675	−	0.998553i
							\(19\)	−2.11169	−	0.874693i	−0.484456	−	0.200668i	0.127068	−	0.991894i	\(-0.459443\pi\)
−0.611524	+	0.791226i	\(0.709443\pi\)
\(23\)	−5.76117	+	3.84949i	−1.20129	+	0.802674i	−0.984814	−	0.173614i	\(-0.944455\pi\)
							−0.216473	+	0.976289i	\(0.569455\pi\)
\(29\)	9.62997	−	1.91552i	1.78824	−	0.355703i	0.813939	−	0.580951i	\(-0.197319\pi\)
							0.974302	+	0.225248i	\(0.0723191\pi\)
\(31\)	2.57416	−	3.85250i	0.462333	−	0.691930i	−0.524909	−	0.851158i	\(-0.675901\pi\)
							0.987242	+	0.159228i	\(0.0509006\pi\)
\(37\)	−2.76408	+	4.13673i	−0.454411	+	0.680074i	−0.985965	−	0.166950i	\(-0.946608\pi\)
							0.531554	+	0.847024i	\(0.321608\pi\)
\(41\)	−1.40631	−	0.279733i	−0.219629	−	0.0436869i	0.0840492	−	0.996462i	\(-0.473215\pi\)
							−0.303678	+	0.952775i	\(0.598215\pi\)
\(43\)	5.65530	−	3.31927i	0.862425	−	0.506184i
							\(47\)	5.85702	−	5.85702i	0.854335	−	0.854335i	−0.136329	−	0.990664i	\(-0.543530\pi\)
0.990664	+	0.136329i	\(0.0435304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.3	✓	496
17.12	odd	16	inner	731.2.s.b.386.60	yes	496
43.42	odd	2	inner	731.2.s.b.214.60	yes	496
731.386	even	16	inner	731.2.s.b.386.3	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.3	✓	496	1.1	even	1	trivial
731.2.s.b.214.60	yes	496	43.42	odd	2	inner
731.2.s.b.386.3	yes	496	731.386	even	16	inner
731.2.s.b.386.60	yes	496	17.12	odd	16	inner