Embedded newform 731.2.be.a.59.11

• Label: 731.2.be.a.59.11
• Level: $731$
• Weight: $2$
• Character: 731.59
• 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.be (of order \(56\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{56})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{56}]$

Embedding invariants

Embedding label			59.11
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67502 - 1.05249i) q^{2} +(-2.57521 - 1.82721i) q^{3} +(0.830209 + 1.72395i) q^{4} +(-0.911231 - 3.16295i) q^{5} +(2.39042 + 5.77099i) q^{6} +(-3.94144 + 1.63260i) q^{7} +(-0.0191731 + 0.170166i) q^{8} +(2.30217 + 6.57923i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67502	−	1.05249i	−1.18442	−	0.744221i	−0.211958	−	0.977279i	\(-0.567984\pi\)
							−0.972463	+	0.233058i	\(0.925127\pi\)
\(3\)	−2.57521	−	1.82721i	−1.48680	−	1.05494i	−0.981383	−	0.192062i	\(-0.938483\pi\)
							−0.505414	−	0.862877i	\(-0.668660\pi\)
\(5\)	−0.911231	−	3.16295i	−0.407515	−	1.41452i	−0.854256	−	0.519852i	\(-0.825987\pi\)
							0.446742	−	0.894663i	\(-0.352584\pi\)
\(7\)	−3.94144	+	1.63260i	−1.48973	+	0.617064i	−0.971257	−	0.238033i	\(-0.923498\pi\)
							−0.518468	+	0.855097i	\(0.673498\pi\)
\(11\)	0.0555322	−	0.988843i	0.0167436	−	0.298147i	−0.978863	−	0.204516i	\(-0.934438\pi\)
							0.995607	−	0.0936318i	\(-0.0298476\pi\)
\(13\)	1.80397	−	1.43862i	0.500331	−	0.399000i	−0.340546	−	0.940228i	\(-0.610612\pi\)
							0.840876	+	0.541228i	\(0.182040\pi\)
\(17\)	2.41249	+	3.34364i	0.585115	+	0.810951i
							\(19\)	2.62985	−	7.51567i	0.603328	−	1.72421i	−0.0826394	−	0.996580i	\(-0.526335\pi\)
0.685967	−	0.727632i	\(-0.259379\pi\)
\(23\)	−0.431127	+	7.67693i	−0.0898963	+	1.60075i	0.551923	+	0.833895i	\(0.313894\pi\)
							−0.641819	+	0.766856i	\(0.721820\pi\)
\(29\)	−1.01069	+	5.94849i	−0.187680	+	1.10461i	0.722801	+	0.691056i	\(0.242854\pi\)
							−0.910481	+	0.413550i	\(0.864289\pi\)
\(31\)	−3.68371	−	0.625888i	−0.661614	−	0.112413i	−0.173012	−	0.984920i	\(-0.555350\pi\)
							−0.488602	+	0.872507i	\(0.662493\pi\)
\(37\)	−1.98278	+	4.78686i	−0.325968	+	0.786955i	0.672916	+	0.739719i	\(0.265041\pi\)
							−0.998884	+	0.0472366i	\(0.984959\pi\)
\(41\)	2.23377	+	3.14820i	0.348856	+	0.491667i	0.952056	−	0.305923i	\(-0.0989651\pi\)
							−0.603200	+	0.797590i	\(0.706108\pi\)
\(43\)	−6.40891	+	1.38775i	−0.977350	+	0.211630i
							\(47\)	−1.74219	−	3.61769i	−0.254124	−	0.527694i	0.734407	−	0.678709i	\(-0.237460\pi\)
−0.988532	+	0.151015i	\(0.951746\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.11	✓	1536
17.15	even	8	inner	731.2.be.a.661.54	yes	1536
43.35	even	7	inner	731.2.be.a.637.54	yes	1536
731.508	even	56	inner	731.2.be.a.508.11	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.11	✓	1536	1.1	even	1	trivial
731.2.be.a.508.11	yes	1536	731.508	even	56	inner
731.2.be.a.637.54	yes	1536	43.35	even	7	inner
731.2.be.a.661.54	yes	1536	17.15	even	8	inner