Embedded newform 845.2.bn.a.392.45

• Label: 845.2.bn.a.392.45
• 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			392.45
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0359046 - 0.0756673i) q^{2} +(-0.0823937 - 0.0760103i) q^{3} +(1.26045 + 1.54378i) q^{4} +(-2.19605 - 0.421157i) q^{5} +(-0.00870981 + 0.00350540i) q^{6} +(1.43323 + 0.415141i) q^{7} +(0.324710 - 0.0800337i) q^{8} +(-0.240389 - 2.97775i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0359046	−	0.0756673i	0.0253884	−	0.0535049i	−0.890360	−	0.455257i	\(-0.849547\pi\)
							0.915748	+	0.401752i	\(0.131599\pi\)
\(3\)	−0.0823937	−	0.0760103i	−0.0475700	−	0.0438846i	0.653921	−	0.756563i	\(-0.273123\pi\)
							−0.701491	+	0.712678i	\(0.747482\pi\)
\(5\)	−2.19605	−	0.421157i	−0.982103	−	0.188347i
							\(7\)	1.43323	+	0.415141i	0.541711	+	0.156909i	0.537782	−	0.843084i	\(-0.319262\pi\)
0.00392899	+	0.999992i	\(0.498749\pi\)
\(11\)	3.49780	+	4.11214i	1.05463	+	1.23986i	0.969913	+	0.243452i	\(0.0782799\pi\)
							0.0847145	+	0.996405i	\(0.473002\pi\)
\(13\)	3.18126	+	1.69694i	0.882322	+	0.470647i
							\(17\)	−1.26290	−	2.29283i	−0.306298	−	0.556093i	0.677467	−	0.735553i	\(-0.263078\pi\)
−0.983765	+	0.179460i	\(0.942565\pi\)
\(19\)	−7.14805	+	1.91531i	−1.63988	+	0.439403i	−0.956753	−	0.290901i	\(-0.906045\pi\)
							−0.683122	+	0.730304i	\(0.739378\pi\)
\(23\)	0.940624	−	3.51046i	0.196134	−	0.731980i	−0.795837	−	0.605511i	\(-0.792969\pi\)
							0.991971	−	0.126469i	\(-0.0403646\pi\)
\(29\)	7.15480	+	3.39499i	1.32861	+	0.630435i	0.954850	−	0.297090i	\(-0.0960160\pi\)
							0.373763	+	0.927524i	\(0.378067\pi\)
\(31\)	−0.427841	−	0.335192i	−0.0768425	−	0.0602023i	0.577563	−	0.816346i	\(-0.304004\pi\)
							−0.654405	+	0.756144i	\(0.727081\pi\)
\(37\)	6.63588	+	8.83076i	1.09093	+	1.45177i	0.880431	+	0.474174i	\(0.157253\pi\)
							0.210501	+	0.977594i	\(0.432490\pi\)
\(41\)	2.20782	+	2.03677i	0.344804	+	0.318090i	0.831546	−	0.555455i	\(-0.187456\pi\)
							−0.486742	+	0.873546i	\(0.661815\pi\)
\(43\)	0.908218	+	6.39995i	0.138502	+	0.975984i	0.930717	+	0.365741i	\(0.119184\pi\)
							−0.792215	+	0.610242i	\(0.791072\pi\)
\(47\)	9.22885	−	3.50004i	1.34617	−	0.510533i	0.426946	−	0.904277i	\(-0.359590\pi\)
							0.919220	+	0.393744i	\(0.128820\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.45	yes	4272
5.3	odd	4		845.2.bi.a.223.45	yes	4272
169.72	odd	156		845.2.bi.a.72.45	✓	4272
845.748	even	156	inner	845.2.bn.a.748.45	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.45	✓	4272	169.72	odd	156
845.2.bi.a.223.45	yes	4272	5.3	odd	4
845.2.bn.a.392.45	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.45	yes	4272	845.748	even	156	inner