Embedded newform 845.2.bn.a.392.55

• Label: 845.2.bn.a.392.55
• Level: $845$
• Weight: $2$
• Character: 845.392
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $4272$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bn (of order \(156\), degree \(48\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(4272\)
Relative dimension:	\(89\) over \(\Q(\zeta_{156})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label			392.55
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.273840 - 0.577106i) q^{2} +(-1.23956 - 1.14353i) q^{3} +(1.00683 + 1.23314i) q^{4} +(0.121711 + 2.23275i) q^{5} +(-0.999378 + 0.402215i) q^{6} +(0.496828 + 0.143908i) q^{7} +(2.22780 - 0.549103i) q^{8} +(-0.0125395 - 0.155330i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{61}{156}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.273840	−	0.577106i	0.193634	−	0.408076i	−0.783292	−	0.621655i	\(-0.786461\pi\)
							0.976926	+	0.213579i	\(0.0685121\pi\)
\(3\)	−1.23956	−	1.14353i	−0.715661	−	0.660215i	0.234450	−	0.972128i	\(-0.424671\pi\)
							−0.950111	+	0.311913i	\(0.899030\pi\)
\(5\)	0.121711	+	2.23275i	0.0544307	+	0.998518i
							\(7\)	0.496828	+	0.143908i	0.187783	+	0.0543921i	0.370777	−	0.928722i	\(-0.379091\pi\)
−0.182993	+	0.983114i	\(0.558579\pi\)
\(11\)	−1.43498	−	1.68701i	−0.432662	−	0.508653i	0.501668	−	0.865060i	\(-0.332720\pi\)
							−0.934331	+	0.356407i	\(0.884002\pi\)
\(13\)	−0.305042	+	3.59262i	−0.0846033	+	0.996415i
							\(17\)	−0.326999	−	0.593675i	−0.0793089	−	0.143987i	0.833300	−	0.552821i	\(-0.186449\pi\)
−0.912609	+	0.408833i	\(0.865936\pi\)
\(19\)	6.42593	−	1.72182i	1.47421	−	0.395013i	0.569837	−	0.821758i	\(-0.307006\pi\)
							0.904372	+	0.426745i	\(0.140340\pi\)
\(23\)	−1.17233	+	4.37520i	−0.244448	+	0.912292i	0.729212	+	0.684287i	\(0.239887\pi\)
							−0.973660	+	0.228004i	\(0.926780\pi\)
\(29\)	6.74662	+	3.20131i	1.25282	+	0.594469i	0.935324	−	0.353792i	\(-0.115108\pi\)
							0.317492	+	0.948261i	\(0.397159\pi\)
\(31\)	6.92358	+	5.42428i	1.24351	+	0.974230i	0.999991	+	0.00433402i	\(0.00137957\pi\)
							0.243522	+	0.969895i	\(0.421697\pi\)
\(37\)	−0.801336	−	1.06638i	−0.131739	−	0.175312i	0.728753	−	0.684776i	\(-0.240100\pi\)
							−0.860492	+	0.509464i	\(0.829844\pi\)
\(41\)	−3.63663	−	3.35488i	−0.567947	−	0.523945i	0.341436	−	0.939905i	\(-0.389087\pi\)
							−0.909383	+	0.415960i	\(0.863446\pi\)
\(43\)	1.81202	+	12.7688i	0.276330	+	1.94722i	0.318007	+	0.948089i	\(0.396987\pi\)
							−0.0416763	+	0.999131i	\(0.513270\pi\)
\(47\)	2.95395	−	1.12029i	0.430879	−	0.163411i	−0.129624	−	0.991563i	\(-0.541377\pi\)
							0.560503	+	0.828152i	\(0.310608\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.55	yes	4272
5.3	odd	4		845.2.bi.a.223.35	yes	4272
169.72	odd	156		845.2.bi.a.72.35	✓	4272
845.748	even	156	inner	845.2.bn.a.748.55	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.35	✓	4272	169.72	odd	156
845.2.bi.a.223.35	yes	4272	5.3	odd	4
845.2.bn.a.392.55	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.55	yes	4272	845.748	even	156	inner