Embedded newform 731.2.be.a.59.8

• Label: 731.2.be.a.59.8
• 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			59.8
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.94177 - 1.22010i) q^{2} +(-1.28164 - 0.909373i) q^{3} +(1.41408 + 2.93637i) q^{4} +(0.0278087 + 0.0965260i) q^{5} +(1.37913 + 3.32952i) q^{6} +(2.68860 - 1.11366i) q^{7} +(0.323299 - 2.86936i) q^{8} +(-0.175193 - 0.500673i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.94177	−	1.22010i	−1.37304	−	0.862739i	−0.375107	−	0.926982i	\(-0.622394\pi\)
							−0.997934	+	0.0642426i	\(0.979537\pi\)
\(3\)	−1.28164	−	0.909373i	−0.739956	−	0.525026i	0.145723	−	0.989325i	\(-0.453449\pi\)
							−0.885679	+	0.464299i	\(0.846306\pi\)
\(5\)	0.0278087	+	0.0965260i	0.0124364	+	0.0431677i	0.966893	−	0.255182i	\(-0.0821353\pi\)
							−0.954457	+	0.298350i	\(0.903564\pi\)
\(7\)	2.68860	−	1.11366i	1.01620	−	0.420923i	0.188485	−	0.982076i	\(-0.439642\pi\)
							0.827712	+	0.561153i	\(0.189642\pi\)
\(11\)	−0.0113738	+	0.202529i	−0.00342933	+	0.0610648i	−0.999675	−	0.0255118i	\(-0.991878\pi\)
							0.996245	+	0.0865766i	\(0.0275927\pi\)
\(13\)	−1.96890	+	1.57015i	−0.546076	+	0.435481i	−0.857271	−	0.514865i	\(-0.827842\pi\)
							0.311195	+	0.950346i	\(0.399271\pi\)
\(17\)	2.40383	+	3.34986i	0.583016	+	0.812461i
							\(19\)	−0.658102	+	1.88075i	−0.150979	+	0.431473i	−0.994387	−	0.105800i	\(-0.966260\pi\)
0.843409	+	0.537273i	\(0.180545\pi\)
\(23\)	−0.357167	+	6.35995i	−0.0744745	+	1.32614i	0.709216	+	0.704991i	\(0.249049\pi\)
							−0.783691	+	0.621151i	\(0.786665\pi\)
\(29\)	−1.17937	+	6.94128i	−0.219004	+	1.28896i	0.636544	+	0.771240i	\(0.280363\pi\)
							−0.855548	+	0.517724i	\(0.826780\pi\)
\(31\)	8.52153	+	1.44787i	1.53051	+	0.260044i	0.870862	−	0.491528i	\(-0.163562\pi\)
							0.659650	+	0.751573i	\(0.270704\pi\)
\(37\)	3.89456	−	9.40229i	0.640261	−	1.54573i	−0.186067	−	0.982537i	\(-0.559574\pi\)
							0.826328	−	0.563189i	\(-0.190426\pi\)
\(41\)	2.30094	+	3.24288i	0.359347	+	0.506452i	0.954926	−	0.296845i	\(-0.0959343\pi\)
							−0.595579	+	0.803297i	\(0.703077\pi\)
\(43\)	−4.29466	+	4.95539i	−0.654929	+	0.755690i
							\(47\)	−3.47399	−	7.21381i	−0.506733	−	1.05224i	−0.984764	−	0.173899i	\(-0.944364\pi\)
0.478031	−	0.878343i	\(-0.341351\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.8	✓	1536
17.15	even	8	inner	731.2.be.a.661.57	yes	1536
43.35	even	7	inner	731.2.be.a.637.57	yes	1536
731.508	even	56	inner	731.2.be.a.508.8	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.8	✓	1536	1.1	even	1	trivial
731.2.be.a.508.8	yes	1536	731.508	even	56	inner
731.2.be.a.637.57	yes	1536	43.35	even	7	inner
731.2.be.a.661.57	yes	1536	17.15	even	8	inner