Embedded newform 845.2.v.a.389.32

• Label: 845.2.v.a.389.32
• Level: $845$
• Weight: $2$
• Character: 845.389
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $1056$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.v (of order \(26\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(1056\)
Relative dimension:	\(88\) over \(\Q(\zeta_{26})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

Embedding invariants

Embedding label			389.32
Character	\(\chi\)	\(=\)	845.389
Dual form			845.2.v.a.454.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.727878 - 0.644844i) q^{2} +(-1.28868 - 0.488731i) q^{3} +(-0.127090 - 1.04668i) q^{4} +(2.03328 - 0.930468i) q^{5} +(0.622845 + 1.18673i) q^{6} +(2.63622 - 3.81922i) q^{7} +(-1.68725 + 2.44441i) q^{8} +(-0.823701 - 0.729735i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1056 q - 106 q^{4} - 13 q^{5} - 26 q^{6} + 62 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{19}{26}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.727878	−	0.644844i	−0.514688	−	0.455973i	0.365355	−	0.930868i	\(-0.380948\pi\)
							−0.880043	+	0.474895i	\(0.842486\pi\)
\(3\)	−1.28868	−	0.488731i	−0.744018	−	0.282169i	−0.0466788	−	0.998910i	\(-0.514864\pi\)
							−0.697339	+	0.716741i	\(0.745633\pi\)
\(5\)	2.03328	−	0.930468i	0.909311	−	0.416118i
							\(7\)	2.63622	−	3.81922i	0.996398	−	1.44353i	0.102779	−	0.994704i	\(-0.467227\pi\)
0.893619	−	0.448827i	\(-0.148158\pi\)
\(11\)	−3.20587	−	3.61868i	−0.966606	−	1.09107i	−0.995805	−	0.0914989i	\(-0.970834\pi\)
							0.0291993	−	0.999574i	\(-0.490704\pi\)
\(13\)	−2.15749	+	2.88881i	−0.598381	+	0.801212i
							\(17\)	3.62586	+	2.50275i	0.879401	+	0.607006i	0.919891	−	0.392173i	\(-0.128277\pi\)
−0.0404906	+	0.999180i	\(0.512892\pi\)
\(19\)		−	6.04293i		−	1.38634i	−0.720772	−	0.693172i	\(-0.756213\pi\)
							0.720772	−	0.693172i	\(-0.243787\pi\)
\(23\)		−	6.18896i		−	1.29049i	−0.763977	−	0.645244i	\(-0.776756\pi\)
							0.763977	−	0.645244i	\(-0.223244\pi\)
\(29\)	6.73020	+	5.96244i	1.24977	+	1.10720i	0.989932	+	0.141544i	\(0.0452068\pi\)
							0.259836	+	0.965653i	\(0.416332\pi\)
\(31\)	−0.754668	−	1.43790i	−0.135542	−	0.258254i	0.808219	−	0.588882i	\(-0.200432\pi\)
							−0.943761	+	0.330628i	\(0.892740\pi\)
\(37\)	−4.72857	+	2.48174i	−0.777372	+	0.407996i	−0.806241	−	0.591587i	\(-0.798502\pi\)
							0.0288695	+	0.999583i	\(0.490809\pi\)
\(41\)	1.27616	+	0.483985i	0.199303	+	0.0755858i	0.452237	−	0.891898i	\(-0.350626\pi\)
							−0.252934	+	0.967484i	\(0.581395\pi\)
\(43\)	−2.13085	+	4.05999i	−0.324951	+	0.619143i	−0.992053	−	0.125824i	\(-0.959842\pi\)
							0.667101	+	0.744967i	\(0.267535\pi\)
\(47\)	0.359715	−	2.96252i	0.0524699	−	0.432128i	−0.942675	−	0.333711i	\(-0.891699\pi\)
							0.995145	−	0.0984171i	\(-0.0313779\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.v.a.389.32	✓	1056
5.4	even	2	inner	845.2.v.a.389.57	yes	1056
169.116	even	26	inner	845.2.v.a.454.57	yes	1056
845.454	even	26	inner	845.2.v.a.454.32	yes	1056

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.v.a.389.32	✓	1056	1.1	even	1	trivial
845.2.v.a.389.57	yes	1056	5.4	even	2	inner
845.2.v.a.454.32	yes	1056	845.454	even	26	inner
845.2.v.a.454.57	yes	1056	169.116	even	26	inner