Embedded newform 731.2.bh.a.13.3

• Label: 731.2.bh.a.13.3
• Level: $731$
• Weight: $2$
• Character: 731.13
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1536$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bh (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.57960 - 0.588776i) q^{2} +(-3.03566 - 0.113587i) q^{3} +(4.50572 + 2.16984i) q^{4} +(1.25804 + 1.70458i) q^{5} +(7.76391 + 2.08033i) q^{6} +(0.317974 + 1.18669i) q^{7} +(-6.20805 - 4.95075i) q^{8} +(6.21075 + 0.465431i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.57960	−	0.588776i	−1.82405	−	0.416328i	−0.833391	−	0.552684i	\(-0.813603\pi\)
							−0.990660	+	0.136357i	\(0.956461\pi\)
\(3\)	−3.03566	−	0.113587i	−1.75264	−	0.0655792i	−0.858350	−	0.513064i	\(-0.828510\pi\)
							−0.894292	+	0.447485i	\(0.852320\pi\)
\(5\)	1.25804	+	1.70458i	0.562613	+	0.762313i	0.989895	−	0.141805i	\(-0.0452907\pi\)
							−0.427282	+	0.904118i	\(0.640529\pi\)
\(7\)	0.317974	+	1.18669i	0.120183	+	0.448528i	0.999622	−	0.0274826i	\(-0.00874909\pi\)
							−0.879440	+	0.476011i	\(0.842082\pi\)
\(11\)	0.141919	−	0.405580i	0.0427901	−	0.122287i	−0.920506	−	0.390729i	\(-0.872223\pi\)
							0.963296	+	0.268442i	\(0.0865087\pi\)
\(13\)	−2.35406	−	5.99806i	−0.652900	−	1.66356i	−0.744381	−	0.667755i	\(-0.767255\pi\)
							0.0914813	−	0.995807i	\(-0.470840\pi\)
\(17\)	3.58577	+	2.03526i	0.869676	+	0.493623i
							\(19\)	3.20698	−	0.240330i	0.735732	−	0.0551355i	0.298407	−	0.954439i	\(-0.403545\pi\)
0.437325	+	0.899303i	\(0.355926\pi\)
\(23\)	−3.84377	+	0.727280i	−0.801481	+	0.151648i	−0.570487	−	0.821307i	\(-0.693245\pi\)
							−0.230994	+	0.972955i	\(0.574198\pi\)
\(29\)	0.361906	+	9.67215i	0.0672043	+	1.79607i	0.471931	+	0.881636i	\(0.343557\pi\)
							−0.404726	+	0.914438i	\(0.632633\pi\)
\(31\)	−7.01213	−	3.70602i	−1.25942	−	0.665620i	−0.301718	−	0.953397i	\(-0.597560\pi\)
							−0.957697	+	0.287777i	\(0.907084\pi\)
\(37\)	0.808601	−	3.01774i	0.132933	−	0.496113i	−0.867065	−	0.498196i	\(-0.833996\pi\)
							0.999998	+	0.00208236i	\(0.000662836\pi\)
\(41\)	−4.30744	−	6.85526i	−0.672710	−	1.07061i	−0.992396	−	0.123088i	\(-0.960720\pi\)
							0.319686	−	0.947523i	\(-0.396422\pi\)
\(43\)	5.46687	−	3.62123i	0.833691	−	0.552232i
							\(47\)	−3.82918	−	1.84404i	−0.558543	−	0.268980i	0.133242	−	0.991084i	\(-0.457461\pi\)
−0.691785	+	0.722103i	\(0.743176\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.3	✓	1536
17.4	even	4	inner	731.2.bh.a.701.62	yes	1536
43.10	even	21	inner	731.2.bh.a.268.62	yes	1536
731.225	even	84	inner	731.2.bh.a.225.3	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.3	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.3	yes	1536	731.225	even	84	inner
731.2.bh.a.268.62	yes	1536	43.10	even	21	inner
731.2.bh.a.701.62	yes	1536	17.4	even	4	inner