Embedded newform 845.2.bn.a.392.1

• Label: 845.2.bn.a.392.1
• 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			392.1
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18690 + 2.50134i) q^{2} +(2.25582 + 2.08105i) q^{3} +(-3.58308 - 4.38848i) q^{4} +(1.92943 + 1.13018i) q^{5} +(-7.88285 + 3.17257i) q^{6} +(-0.850531 - 0.246359i) q^{7} +(9.85344 - 2.42865i) q^{8} +(0.516554 + 6.39867i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18690	+	2.50134i	−0.839266	+	1.76872i	−0.234878	+	0.972025i	\(0.575469\pi\)
							−0.604388	+	0.796690i	\(0.706582\pi\)
\(3\)	2.25582	+	2.08105i	1.30240	+	1.20149i	0.965634	+	0.259906i	\(0.0836915\pi\)
							0.336764	+	0.941589i	\(0.390668\pi\)
\(5\)	1.92943	+	1.13018i	0.862867	+	0.505431i
							\(7\)	−0.850531	−	0.246359i	−0.321470	−	0.0931150i	0.113559	−	0.993531i	\(-0.463775\pi\)
−0.435029	+	0.900416i	\(0.643262\pi\)
\(11\)	2.54995	+	2.99781i	0.768838	+	0.903874i	0.997574	−	0.0696149i	\(-0.0221771\pi\)
							−0.228736	+	0.973489i	\(0.573459\pi\)
\(13\)	−3.28107	−	1.49485i	−0.910005	−	0.414597i
							\(17\)	2.27548	+	4.13120i	0.551885	+	1.00196i	0.994448	+	0.105225i	\(0.0335564\pi\)
−0.442563	+	0.896737i	\(0.645931\pi\)
\(19\)	1.16779	−	0.312909i	0.267910	−	0.0717862i	−0.122363	−	0.992485i	\(-0.539047\pi\)
							0.390273	+	0.920699i	\(0.372381\pi\)
\(23\)	0.841054	−	3.13886i	0.175372	−	0.654497i	−0.821116	−	0.570761i	\(-0.806648\pi\)
							0.996488	−	0.0837355i	\(-0.0266851\pi\)
\(29\)	5.98943	+	2.84202i	1.11221	+	0.527750i	0.893961	−	0.448144i	\(-0.147915\pi\)
							0.218247	+	0.975893i	\(0.429966\pi\)
\(31\)	−2.96838	−	2.32558i	−0.533137	−	0.417686i	0.313713	−	0.949518i	\(-0.398427\pi\)
							−0.846850	+	0.531832i	\(0.821504\pi\)
\(37\)	−3.97929	−	5.29547i	−0.654191	−	0.870570i	0.343454	−	0.939170i	\(-0.388403\pi\)
							−0.997644	+	0.0686000i	\(0.978147\pi\)
\(41\)	−2.95386	−	2.72501i	−0.461315	−	0.425574i	0.413150	−	0.910663i	\(-0.364428\pi\)
							−0.874465	+	0.485088i	\(0.838787\pi\)
\(43\)	−1.30075	−	9.16599i	−0.198362	−	1.39780i	−0.800464	−	0.599381i	\(-0.795413\pi\)
							0.602101	−	0.798420i	\(-0.294330\pi\)
\(47\)	−4.34873	+	1.64926i	−0.634327	+	0.240569i	−0.650745	−	0.759297i	\(-0.725543\pi\)
							0.0164173	+	0.999865i	\(0.494774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.1	yes	4272
5.3	odd	4		845.2.bi.a.223.89	yes	4272
169.72	odd	156		845.2.bi.a.72.89	✓	4272
845.748	even	156	inner	845.2.bn.a.748.1	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.89	✓	4272	169.72	odd	156
845.2.bi.a.223.89	yes	4272	5.3	odd	4
845.2.bn.a.392.1	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.1	yes	4272	845.748	even	156	inner