Embedded newform 731.2.be.a.59.18

• Label: 731.2.be.a.59.18
• 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			59.18
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33188 - 0.836875i) q^{2} +(-0.817319 - 0.579919i) q^{3} +(0.205773 + 0.427293i) q^{4} +(1.09966 + 3.81700i) q^{5} +(0.603251 + 1.45638i) q^{6} +(0.267978 - 0.111000i) q^{7} +(-0.268710 + 2.38487i) q^{8} +(-0.659132 - 1.88369i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33188	−	0.836875i	−0.941780	−	0.591760i	−0.0287214	−	0.999587i	\(-0.509144\pi\)
							−0.913059	+	0.407828i	\(0.866286\pi\)
\(3\)	−0.817319	−	0.579919i	−0.471880	−	0.334816i	0.317987	−	0.948095i	\(-0.396993\pi\)
							−0.789867	+	0.613279i	\(0.789850\pi\)
\(5\)	1.09966	+	3.81700i	0.491782	+	1.70701i	0.687352	+	0.726324i	\(0.258773\pi\)
							−0.195570	+	0.980690i	\(0.562656\pi\)
\(7\)	0.267978	−	0.111000i	0.101286	−	0.0419542i	−0.331465	−	0.943468i	\(-0.607543\pi\)
							0.432751	+	0.901513i	\(0.357543\pi\)
\(11\)	0.202352	−	3.60322i	0.0610115	−	1.08641i	−0.807220	−	0.590251i	\(-0.799029\pi\)
							0.868231	−	0.496160i	\(-0.165257\pi\)
\(13\)	−1.11159	+	0.886460i	−0.308299	+	0.245860i	−0.765400	−	0.643555i	\(-0.777459\pi\)
							0.457102	+	0.889414i	\(0.348888\pi\)
\(17\)	3.05994	+	2.76347i	0.742144	+	0.670240i
							\(19\)	−0.901889	+	2.57745i	−0.206908	+	0.591308i	−0.999838	−	0.0180044i	\(-0.994269\pi\)
0.792930	+	0.609312i	\(0.208554\pi\)
\(23\)	0.0382056	−	0.680313i	0.00796641	−	0.141855i	−0.991921	−	0.126856i	\(-0.959511\pi\)
							0.999888	−	0.0149988i	\(-0.00477444\pi\)
\(29\)	−0.242839	+	1.42925i	−0.0450940	+	0.265404i	−0.999446	−	0.0332797i	\(-0.989405\pi\)
							0.954352	+	0.298684i	\(0.0965476\pi\)
\(31\)	−7.20330	−	1.22389i	−1.29375	−	0.219817i	−0.520470	−	0.853880i	\(-0.674243\pi\)
							−0.773282	+	0.634063i	\(0.781386\pi\)
\(37\)	−2.82605	+	6.82268i	−0.464599	+	1.12164i	0.501889	+	0.864932i	\(0.332639\pi\)
							−0.966488	+	0.256710i	\(0.917361\pi\)
\(41\)	−0.0292854	−	0.0412740i	−0.00457362	−	0.00644591i	0.813268	−	0.581890i	\(-0.197686\pi\)
							−0.817841	+	0.575444i	\(0.804829\pi\)
\(43\)	5.72910	+	3.19021i	0.873679	+	0.486503i
							\(47\)	3.18414	+	6.61194i	0.464455	+	0.964450i	0.993282	+	0.115721i	\(0.0369177\pi\)
−0.528827	+	0.848730i	\(0.677368\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.18	✓	1536
17.15	even	8	inner	731.2.be.a.661.47	yes	1536
43.35	even	7	inner	731.2.be.a.637.47	yes	1536
731.508	even	56	inner	731.2.be.a.508.18	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.18	✓	1536	1.1	even	1	trivial
731.2.be.a.508.18	yes	1536	731.508	even	56	inner
731.2.be.a.637.47	yes	1536	43.35	even	7	inner
731.2.be.a.661.47	yes	1536	17.15	even	8	inner