Embedded newform 845.2.bn.a.392.48

• Label: 845.2.bn.a.392.48
• 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			392.48
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0992364 - 0.209136i) q^{2} +(2.12311 + 1.95862i) q^{3} +(1.23100 + 1.50770i) q^{4} +(-1.98137 + 1.03643i) q^{5} +(0.620309 - 0.249653i) q^{6} +(-1.10978 - 0.321452i) q^{7} +(0.886994 - 0.218624i) q^{8} +(0.429997 + 5.32647i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0992364	−	0.209136i	0.0701707	−	0.147882i	−0.865334	−	0.501195i	\(-0.832894\pi\)
							0.935505	+	0.353314i	\(0.114945\pi\)
\(3\)	2.12311	+	1.95862i	1.22578	+	1.13081i	0.987166	+	0.159696i	\(0.0510513\pi\)
							0.238612	+	0.971115i	\(0.423308\pi\)
\(5\)	−1.98137	+	1.03643i	−0.886095	+	0.463504i
							\(7\)	−1.10978	−	0.321452i	−0.419458	−	0.121498i	0.0617751	−	0.998090i	\(-0.480324\pi\)
−0.481233	+	0.876593i	\(0.659811\pi\)
\(11\)	−2.06059	−	2.42250i	−0.621292	−	0.730413i	0.357808	−	0.933795i	\(-0.383524\pi\)
							−0.979099	+	0.203383i	\(0.934806\pi\)
\(13\)	−3.57721	−	0.451225i	−0.992138	−	0.125147i
							\(17\)	2.96174	+	5.37712i	0.718327	+	1.30414i	0.944002	+	0.329940i	\(0.107028\pi\)
−0.225675	+	0.974203i	\(0.572459\pi\)
\(19\)	0.0262356	−	0.00702981i	0.00601886	−	0.00161275i	−0.255808	−	0.966728i	\(-0.582342\pi\)
							0.261827	+	0.965115i	\(0.415675\pi\)
\(23\)	−0.197674	+	0.737730i	−0.0412179	+	0.153827i	−0.983468	−	0.181084i	\(-0.942040\pi\)
							0.942250	+	0.334911i	\(0.108706\pi\)
\(29\)	4.20855	+	1.99698i	0.781508	+	0.370830i	0.777275	−	0.629161i	\(-0.216601\pi\)
							0.00423313	+	0.999991i	\(0.498653\pi\)
\(31\)	−2.79474	−	2.18954i	−0.501950	−	0.393253i	0.333559	−	0.942729i	\(-0.391750\pi\)
							−0.835509	+	0.549476i	\(0.814827\pi\)
\(37\)	4.78671	+	6.36995i	0.786930	+	1.04721i	0.997571	+	0.0696501i	\(0.0221883\pi\)
							−0.210642	+	0.977563i	\(0.567555\pi\)
\(41\)	8.52657	+	7.86598i	1.33163	+	1.22846i	0.953022	+	0.302901i	\(0.0979550\pi\)
							0.378605	+	0.925558i	\(0.376404\pi\)
\(43\)	−0.312673	−	2.20332i	−0.0476822	−	0.336002i	−0.999564	−	0.0295363i	\(-0.990597\pi\)
							0.951882	−	0.306466i	\(-0.0991466\pi\)
\(47\)	4.82708	−	1.83067i	0.704102	−	0.267031i	0.0235419	−	0.999723i	\(-0.492506\pi\)
							0.680560	+	0.732692i	\(0.261736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.48	yes	4272
5.3	odd	4		845.2.bi.a.223.42	yes	4272
169.72	odd	156		845.2.bi.a.72.42	✓	4272
845.748	even	156	inner	845.2.bn.a.748.48	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.42	✓	4272	169.72	odd	156
845.2.bi.a.223.42	yes	4272	5.3	odd	4
845.2.bn.a.392.48	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.48	yes	4272	845.748	even	156	inner