Embedded newform 731.2.be.a.59.5

• Label: 731.2.be.a.59.5
• 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			59.5
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.07661 - 1.30482i) q^{2} +(0.254844 + 0.180822i) q^{3} +(1.74198 + 3.61727i) q^{4} +(0.889546 + 3.08768i) q^{5} +(-0.293272 - 0.708022i) q^{6} +(4.10843 - 1.70177i) q^{7} +(0.553273 - 4.91043i) q^{8} +(-0.958588 - 2.73949i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.07661	−	1.30482i	−1.46838	−	0.922647i	−0.999218	−	0.0395379i	\(-0.987411\pi\)
							−0.469167	−	0.883110i	\(-0.655446\pi\)
\(3\)	0.254844	+	0.180822i	0.147134	+	0.104397i	0.649879	−	0.760037i	\(-0.274819\pi\)
							−0.502745	+	0.864435i	\(0.667677\pi\)
\(5\)	0.889546	+	3.08768i	0.397817	+	1.38085i	0.867245	+	0.497882i	\(0.165889\pi\)
							−0.469428	+	0.882971i	\(0.655540\pi\)
\(7\)	4.10843	−	1.70177i	1.55284	−	0.643207i	0.569012	−	0.822329i	\(-0.307326\pi\)
							0.983827	+	0.179122i	\(0.0573258\pi\)
\(11\)	−0.0145477	+	0.259046i	−0.00438631	+	0.0781054i	−0.999856	−	0.0169748i	\(-0.994596\pi\)
							0.995470	+	0.0950802i	\(0.0303108\pi\)
\(13\)	1.05002	−	0.837361i	0.291222	−	0.232242i	−0.466970	−	0.884273i	\(-0.654654\pi\)
							0.758192	+	0.652031i	\(0.226083\pi\)
\(17\)	−1.28210	−	3.91870i	−0.310956	−	0.950424i
							\(19\)	0.432933	−	1.23725i	0.0993215	−	0.283845i	−0.883492	−	0.468447i	\(-0.844814\pi\)
0.982813	+	0.184602i	\(0.0590996\pi\)
\(23\)	−0.0617122	+	1.09889i	−0.0128679	+	0.229134i	0.985398	+	0.170267i	\(0.0544630\pi\)
							−0.998266	+	0.0588669i	\(0.981251\pi\)
\(29\)	1.72047	−	10.1260i	0.319484	−	1.88035i	−0.136929	−	0.990581i	\(-0.543723\pi\)
							0.456413	−	0.889768i	\(-0.349134\pi\)
\(31\)	7.01551	+	1.19198i	1.26002	+	0.214086i	0.758853	−	0.651262i	\(-0.225760\pi\)
							0.501170	+	0.865349i	\(0.332903\pi\)
\(37\)	−1.79365	+	4.33026i	−0.294875	+	0.711891i	0.705121	+	0.709087i	\(0.250893\pi\)
							−0.999996	+	0.00280408i	\(0.999107\pi\)
\(41\)	4.22398	+	5.95314i	0.659675	+	0.929725i	0.999944	−	0.0106035i	\(-0.00337527\pi\)
							−0.340269	+	0.940328i	\(0.610518\pi\)
\(43\)	3.02100	−	5.82010i	0.460698	−	0.887557i
							\(47\)	1.50454	+	3.12420i	0.219459	+	0.455711i	0.981410	−	0.191923i	\(-0.0614725\pi\)
−0.761951	+	0.647635i	\(0.775758\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.5	✓	1536
17.15	even	8	inner	731.2.be.a.661.60	yes	1536
43.35	even	7	inner	731.2.be.a.637.60	yes	1536
731.508	even	56	inner	731.2.be.a.508.5	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.5	✓	1536	1.1	even	1	trivial
731.2.be.a.508.5	yes	1536	731.508	even	56	inner
731.2.be.a.637.60	yes	1536	43.35	even	7	inner
731.2.be.a.661.60	yes	1536	17.15	even	8	inner