Embedded newform 731.2.s.b.214.19

• Label: 731.2.s.b.214.19
• 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			214.19
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25875 - 0.521390i) q^{2} +(1.34853 - 2.01822i) q^{3} +(-0.101617 - 0.101617i) q^{4} +(0.174697 + 0.878263i) q^{5} +(-2.74974 + 1.83732i) q^{6} +(-0.443684 + 2.23055i) q^{7} +(1.11771 + 2.69839i) q^{8} +(-1.10663 - 2.67164i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25875	−	0.521390i	−0.890069	−	0.368678i	−0.109675	−	0.993967i	\(-0.534981\pi\)
							−0.780393	+	0.625289i	\(0.784981\pi\)
\(3\)	1.34853	−	2.01822i	0.778576	−	1.16522i	−0.203929	−	0.978986i	\(-0.565371\pi\)
							0.982505	−	0.186235i	\(-0.0596287\pi\)
\(5\)	0.174697	+	0.878263i	0.0781270	+	0.392771i	0.999986	+	0.00524605i	\(0.00166988\pi\)
							−0.921859	+	0.387525i	\(0.873330\pi\)
\(7\)	−0.443684	+	2.23055i	−0.167697	+	0.843069i	0.801730	+	0.597687i	\(0.203913\pi\)
							−0.969427	+	0.245382i	\(0.921087\pi\)
\(11\)	1.83259	+	2.74267i	0.552547	+	0.826945i	0.997649	−	0.0685350i	\(-0.0218325\pi\)
							−0.445102	+	0.895480i	\(0.646832\pi\)
\(13\)	4.43469	−	4.43469i	1.22996	−	1.22996i	0.265984	−	0.963977i	\(-0.414303\pi\)
							0.963977	−	0.265984i	\(-0.0856968\pi\)
\(17\)	−0.758169	+	4.05280i	−0.183883	+	0.982948i
							\(19\)	−4.06244	−	1.68272i	−0.931987	−	0.386042i	−0.135555	−	0.990770i	\(-0.543282\pi\)
−0.796432	+	0.604728i	\(0.793282\pi\)
\(23\)	4.38129	−	2.92748i	0.913562	−	0.610422i	−0.00744362	−	0.999972i	\(-0.502369\pi\)
							0.921005	+	0.389550i	\(0.127369\pi\)
\(29\)	8.46919	−	1.68463i	1.57269	−	0.312827i	0.669747	−	0.742589i	\(-0.266402\pi\)
							0.902942	+	0.429762i	\(0.141402\pi\)
\(31\)	0.938922	−	1.40520i	0.168635	−	0.252381i	−0.737520	−	0.675325i	\(-0.764003\pi\)
							0.906156	+	0.422944i	\(0.139003\pi\)
\(37\)	−4.23591	+	6.33949i	−0.696379	+	1.04221i	0.299725	+	0.954026i	\(0.403105\pi\)
							−0.996105	+	0.0881798i	\(0.971895\pi\)
\(41\)	−7.79827	−	1.55117i	−1.21789	−	0.242252i	−0.456009	−	0.889975i	\(-0.650722\pi\)
							−0.761876	+	0.647723i	\(0.775722\pi\)
\(43\)	−1.93378	+	6.26582i	−0.294898	+	0.955529i
							\(47\)	5.57783	−	5.57783i	0.813610	−	0.813610i	−0.171563	−	0.985173i	\(-0.554882\pi\)
0.985173	+	0.171563i	\(0.0548817\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.19	✓	496
17.12	odd	16	inner	731.2.s.b.386.44	yes	496
43.42	odd	2	inner	731.2.s.b.214.44	yes	496
731.386	even	16	inner	731.2.s.b.386.19	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.19	✓	496	1.1	even	1	trivial
731.2.s.b.214.44	yes	496	43.42	odd	2	inner
731.2.s.b.386.19	yes	496	731.386	even	16	inner
731.2.s.b.386.44	yes	496	17.12	odd	16	inner