Embedded newform 731.2.s.b.214.20

• Label: 731.2.s.b.214.20
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• 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			214.20
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16026 - 0.480596i) q^{2} +(-0.484485 + 0.725082i) q^{3} +(-0.298982 - 0.298982i) q^{4} +(0.671308 + 3.37489i) q^{5} +(0.910600 - 0.608443i) q^{6} +(-0.264895 + 1.33172i) q^{7} +(1.16440 + 2.81111i) q^{8} +(0.857031 + 2.06906i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16026	−	0.480596i	−0.820428	−	0.339832i	−0.0673217	−	0.997731i	\(-0.521445\pi\)
							−0.753106	+	0.657899i	\(0.771445\pi\)
\(3\)	−0.484485	+	0.725082i	−0.279717	+	0.418627i	−0.944551	−	0.328365i	\(-0.893502\pi\)
							0.664834	+	0.746991i	\(0.268502\pi\)
\(5\)	0.671308	+	3.37489i	0.300218	+	1.50930i	0.776563	+	0.630040i	\(0.216961\pi\)
							−0.476345	+	0.879259i	\(0.658039\pi\)
\(7\)	−0.264895	+	1.33172i	−0.100121	+	0.503341i	0.897886	+	0.440229i	\(0.145103\pi\)
							−0.998006	+	0.0631125i	\(0.979897\pi\)
\(11\)	0.514526	+	0.770042i	0.155135	+	0.232176i	0.900893	−	0.434040i	\(-0.142913\pi\)
							−0.745758	+	0.666217i	\(0.767913\pi\)
\(13\)	−3.50029	+	3.50029i	−0.970805	+	0.970805i	−0.999586	−	0.0287804i	\(-0.990838\pi\)
							0.0287804	+	0.999586i	\(0.490838\pi\)
\(17\)	−2.33228	−	3.40007i	−0.565662	−	0.824637i
							\(19\)	2.29491	+	0.950581i	0.526487	+	0.218078i	0.630064	−	0.776543i	\(-0.283029\pi\)
−0.103577	+	0.994621i	\(0.533029\pi\)
\(23\)	3.31901	−	2.21769i	0.692062	−	0.462421i	−0.159142	−	0.987256i	\(-0.550873\pi\)
							0.851204	+	0.524835i	\(0.175873\pi\)
\(29\)	1.27118	−	0.252854i	0.236053	−	0.0469539i	−0.0756461	−	0.997135i	\(-0.524102\pi\)
							0.311699	+	0.950181i	\(0.399102\pi\)
\(31\)	3.90950	−	5.85098i	0.702166	−	1.05087i	−0.293325	−	0.956013i	\(-0.594762\pi\)
							0.995491	−	0.0948534i	\(-0.0302382\pi\)
\(37\)	−3.73831	+	5.59478i	−0.614574	+	0.919775i	−0.999996	−	0.00296532i	\(-0.999056\pi\)
							0.385421	+	0.922741i	\(0.374056\pi\)
\(41\)	−7.81212	−	1.55393i	−1.22005	−	0.242683i	−0.457261	−	0.889332i	\(-0.651170\pi\)
							−0.762787	+	0.646650i	\(0.776170\pi\)
\(43\)	−2.48186	+	6.06963i	−0.378480	+	0.925610i
							\(47\)	−4.98220	+	4.98220i	−0.726729	+	0.726729i	−0.969967	−	0.243238i	\(-0.921790\pi\)
0.243238	+	0.969967i	\(0.421790\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.20	✓	496
17.12	odd	16	inner	731.2.s.b.386.43	yes	496
43.42	odd	2	inner	731.2.s.b.214.43	yes	496
731.386	even	16	inner	731.2.s.b.386.20	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.20	✓	496	1.1	even	1	trivial
731.2.s.b.214.43	yes	496	43.42	odd	2	inner
731.2.s.b.386.20	yes	496	731.386	even	16	inner
731.2.s.b.386.43	yes	496	17.12	odd	16	inner