Embedded newform 731.2.s.b.214.2

• Label: 731.2.s.b.214.2
• 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			214.2
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49245 - 1.03241i) q^{2} +(-1.39302 + 2.08480i) q^{3} +(3.73222 + 3.73222i) q^{4} +(0.435881 + 2.19132i) q^{5} +(5.62439 - 3.75810i) q^{6} +(-0.139368 + 0.700650i) q^{7} +(-3.38438 - 8.17063i) q^{8} +(-1.25784 - 3.03670i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49245	−	1.03241i	−1.76243	−	0.730021i	−0.996165	−	0.0874915i	\(-0.972115\pi\)
							−0.766261	−	0.642529i	\(-0.777885\pi\)
\(3\)	−1.39302	+	2.08480i	−0.804260	+	1.20366i	0.171579	+	0.985170i	\(0.445113\pi\)
							−0.975839	+	0.218490i	\(0.929887\pi\)
\(5\)	0.435881	+	2.19132i	0.194932	+	0.979990i	0.947081	+	0.320996i	\(0.104018\pi\)
							−0.752148	+	0.658994i	\(0.770982\pi\)
\(7\)	−0.139368	+	0.700650i	−0.0526761	+	0.264821i	−0.998144	−	0.0608939i	\(-0.980605\pi\)
							0.945468	+	0.325715i	\(0.105605\pi\)
\(11\)	0.171726	+	0.257006i	0.0517772	+	0.0774901i	0.856455	−	0.516222i	\(-0.172662\pi\)
							−0.804678	+	0.593712i	\(0.797662\pi\)
\(13\)	−3.53410	+	3.53410i	−0.980184	+	0.980184i	−0.999807	−	0.0196233i	\(-0.993753\pi\)
							0.0196233	+	0.999807i	\(0.493753\pi\)
\(17\)	−3.69018	+	1.83918i	−0.894999	+	0.446068i
							\(19\)	5.12923	+	2.12460i	1.17673	+	0.487416i	0.883410	−	0.468600i	\(-0.155241\pi\)
0.293315	+	0.956016i	\(0.405241\pi\)
\(23\)	−0.374192	+	0.250027i	−0.0780244	+	0.0521343i	−0.593971	−	0.804487i	\(-0.702440\pi\)
							0.515946	+	0.856621i	\(0.327440\pi\)
\(29\)	−6.06876	+	1.20715i	−1.12694	+	0.224162i	−0.723151	−	0.690690i	\(-0.757307\pi\)
							−0.403789	+	0.914852i	\(0.632307\pi\)
\(31\)	−2.81287	+	4.20976i	−0.505206	+	0.756095i	−0.993158	−	0.116776i	\(-0.962744\pi\)
							0.487952	+	0.872870i	\(0.337744\pi\)
\(37\)	1.91297	−	2.86296i	0.314490	−	0.470667i	−0.640226	−	0.768187i	\(-0.721159\pi\)
							0.954716	+	0.297519i	\(0.0961593\pi\)
\(41\)	4.07595	+	0.810757i	0.636557	+	0.126619i	0.502811	−	0.864397i	\(-0.332299\pi\)
							0.133746	+	0.991016i	\(0.457299\pi\)
\(43\)	−4.97338	+	4.27382i	−0.758433	+	0.651751i
							\(47\)	4.81652	−	4.81652i	0.702561	−	0.702561i	−0.262398	−	0.964960i	\(-0.584513\pi\)
0.964960	+	0.262398i	\(0.0845135\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.2	✓	496
17.12	odd	16	inner	731.2.s.b.386.61	yes	496
43.42	odd	2	inner	731.2.s.b.214.61	yes	496
731.386	even	16	inner	731.2.s.b.386.2	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.2	✓	496	1.1	even	1	trivial
731.2.s.b.214.61	yes	496	43.42	odd	2	inner
731.2.s.b.386.2	yes	496	731.386	even	16	inner
731.2.s.b.386.61	yes	496	17.12	odd	16	inner