Embedded newform 731.2.s.b.214.14

• Label: 731.2.s.b.214.14
• 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			214.14
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.55591 - 0.644478i) q^{2} +(-0.529491 + 0.792439i) q^{3} +(0.591285 + 0.591285i) q^{4} +(-0.134545 - 0.676403i) q^{5} +(1.33455 - 0.891717i) q^{6} +(-0.834241 + 4.19401i) q^{7} +(0.750042 + 1.81076i) q^{8} +(0.800452 + 1.93246i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.55591	−	0.644478i	−1.10019	−	0.455715i	−0.242643	−	0.970116i	\(-0.578014\pi\)
							−0.857550	+	0.514401i	\(0.828014\pi\)
\(3\)	−0.529491	+	0.792439i	−0.305702	+	0.457515i	−0.952233	−	0.305373i	\(-0.901219\pi\)
							0.646531	+	0.762887i	\(0.276219\pi\)
\(5\)	−0.134545	−	0.676403i	−0.0601704	−	0.302497i	0.938968	−	0.344005i	\(-0.111784\pi\)
							−0.999138	+	0.0415082i	\(0.986784\pi\)
\(7\)	−0.834241	+	4.19401i	−0.315313	+	1.58519i	0.420042	+	0.907505i	\(0.362015\pi\)
							−0.735355	+	0.677682i	\(0.762985\pi\)
\(11\)	0.818950	+	1.22564i	0.246923	+	0.369546i	0.934140	−	0.356906i	\(-0.116168\pi\)
							−0.687218	+	0.726451i	\(0.741168\pi\)
\(13\)	2.31054	−	2.31054i	0.640827	−	0.640827i	−0.309931	−	0.950759i	\(-0.600306\pi\)
							0.950759	+	0.309931i	\(0.100306\pi\)
\(17\)	−3.88076	−	1.39273i	−0.941222	−	0.337788i
							\(19\)	−0.747234	−	0.309514i	−0.171427	−	0.0710075i	0.295319	−	0.955399i	\(-0.404574\pi\)
−0.466747	+	0.884391i	\(0.654574\pi\)
\(23\)	−4.63130	+	3.09454i	−0.965693	+	0.645256i	−0.935140	−	0.354278i	\(-0.884727\pi\)
							−0.0305528	+	0.999533i	\(0.509727\pi\)
\(29\)	−3.27896	+	0.652225i	−0.608887	+	0.121115i	−0.489897	−	0.871780i	\(-0.662966\pi\)
							−0.118990	+	0.992895i	\(0.537966\pi\)
\(31\)	−0.756444	+	1.13210i	−0.135861	+	0.203331i	−0.893162	−	0.449736i	\(-0.851518\pi\)
							0.757300	+	0.653067i	\(0.226518\pi\)
\(37\)	1.95777	−	2.93001i	0.321856	−	0.481691i	−0.634896	−	0.772598i	\(-0.718957\pi\)
							0.956751	+	0.290907i	\(0.0939569\pi\)
\(41\)	10.4327	+	2.07519i	1.62931	+	0.324090i	0.923291	−	0.384100i	\(-0.125488\pi\)
							0.706021	+	0.708191i	\(0.250488\pi\)
\(43\)	−6.14889	+	2.27841i	−0.937697	+	0.347454i
							\(47\)	−5.26809	+	5.26809i	−0.768430	+	0.768430i	−0.977830	−	0.209400i	\(-0.932849\pi\)
0.209400	+	0.977830i	\(0.432849\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.14	✓	496
17.12	odd	16	inner	731.2.s.b.386.49	yes	496
43.42	odd	2	inner	731.2.s.b.214.49	yes	496
731.386	even	16	inner	731.2.s.b.386.14	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.14	✓	496	1.1	even	1	trivial
731.2.s.b.214.49	yes	496	43.42	odd	2	inner
731.2.s.b.386.14	yes	496	731.386	even	16	inner
731.2.s.b.386.49	yes	496	17.12	odd	16	inner