Embedded newform 731.2.s.b.214.7

• Label: 731.2.s.b.214.7
• 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			214.7
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19579 - 0.909528i) q^{2} +(-1.68196 + 2.51724i) q^{3} +(2.58006 + 2.58006i) q^{4} +(-0.588645 - 2.95932i) q^{5} +(5.98274 - 3.99754i) q^{6} +(0.916876 - 4.60945i) q^{7} +(-1.49959 - 3.62032i) q^{8} +(-2.35943 - 5.69616i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19579	−	0.909528i	−1.55266	−	0.643133i	−0.568866	−	0.822430i	\(-0.692618\pi\)
							−0.983795	+	0.179297i	\(0.942618\pi\)
\(3\)	−1.68196	+	2.51724i	−0.971082	+	1.45333i	−0.0814346	+	0.996679i	\(0.525950\pi\)
							−0.889647	+	0.456648i	\(0.849050\pi\)
\(5\)	−0.588645	−	2.95932i	−0.263250	−	1.32345i	−0.855546	−	0.517726i	\(-0.826778\pi\)
							0.592296	−	0.805720i	\(-0.298222\pi\)
\(7\)	0.916876	−	4.60945i	0.346547	−	1.74221i	−0.277413	−	0.960751i	\(-0.589477\pi\)
							0.623959	−	0.781457i	\(-0.285523\pi\)
\(11\)	−2.47081	−	3.69783i	−0.744977	−	1.11494i	−0.989393	−	0.145266i	\(-0.953596\pi\)
							0.244416	−	0.969671i	\(-0.421404\pi\)
\(13\)	3.26601	−	3.26601i	0.905828	−	0.905828i	−0.0901042	−	0.995932i	\(-0.528720\pi\)
							0.995932	+	0.0901042i	\(0.0287200\pi\)
\(17\)	−3.84717	+	1.48300i	−0.933075	+	0.359682i
							\(19\)	−0.528504	−	0.218913i	−0.121247	−	0.0502222i	0.321235	−	0.946999i	\(-0.395902\pi\)
−0.442482	+	0.896777i	\(0.645902\pi\)
\(23\)	−1.26411	+	0.844652i	−0.263585	+	0.176122i	−0.680339	−	0.732897i	\(-0.738168\pi\)
							0.416754	+	0.909019i	\(0.363168\pi\)
\(29\)	2.36143	−	0.469717i	0.438506	−	0.0872243i	0.0291000	−	0.999577i	\(-0.490736\pi\)
							0.409406	+	0.912352i	\(0.365736\pi\)
\(31\)	3.45211	−	5.16645i	0.620018	−	0.927922i	−0.379978	−	0.924995i	\(-0.624069\pi\)
							0.999996	−	0.00292638i	\(-0.000931497\pi\)
\(37\)	−1.63578	+	2.44812i	−0.268920	+	0.402468i	−0.941211	−	0.337820i	\(-0.890311\pi\)
							0.672290	+	0.740287i	\(0.265311\pi\)
\(41\)	3.86212	+	0.768223i	0.603161	+	0.119976i	0.487219	−	0.873280i	\(-0.338011\pi\)
							0.115942	+	0.993256i	\(0.463011\pi\)
\(43\)	−6.55735	−	0.0350179i	−0.999986	−	0.00534019i
							\(47\)	−6.90218	+	6.90218i	−1.00679	+	1.00679i	−0.00680943	+	0.999977i	\(0.502168\pi\)
−0.999977	+	0.00680943i	\(0.997832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.7	✓	496
17.12	odd	16	inner	731.2.s.b.386.56	yes	496
43.42	odd	2	inner	731.2.s.b.214.56	yes	496
731.386	even	16	inner	731.2.s.b.386.7	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.7	✓	496	1.1	even	1	trivial
731.2.s.b.214.56	yes	496	43.42	odd	2	inner
731.2.s.b.386.7	yes	496	731.386	even	16	inner
731.2.s.b.386.56	yes	496	17.12	odd	16	inner