Embedded newform 845.2.bn.a.392.23

• Label: 845.2.bn.a.392.23
• 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			392.23
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.733381 + 1.54557i) q^{2} +(2.11210 + 1.94846i) q^{3} +(-0.586040 - 0.717768i) q^{4} +(0.0766335 + 2.23475i) q^{5} +(-4.56046 + 1.83543i) q^{6} +(4.21224 + 1.22009i) q^{7} +(-1.78290 + 0.439447i) q^{8} +(0.423051 + 5.24043i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.733381	+	1.54557i	−0.518578	+	1.09288i	0.460173	+	0.887829i	\(0.347787\pi\)
							−0.978751	+	0.205052i	\(0.934264\pi\)
\(3\)	2.11210	+	1.94846i	1.21942	+	1.12495i	0.988380	+	0.152002i	\(0.0485721\pi\)
							0.231041	+	0.972944i	\(0.425787\pi\)
\(5\)	0.0766335	+	2.23475i	0.0342715	+	0.999413i
							\(7\)	4.21224	+	1.22009i	1.59208	+	0.461151i	0.951731	−	0.306935i	\(-0.0993033\pi\)
0.640347	+	0.768086i	\(0.278791\pi\)
\(11\)	−2.29406	−	2.69698i	−0.691685	−	0.813169i	0.298281	−	0.954478i	\(-0.403587\pi\)
							−0.989966	+	0.141309i	\(0.954869\pi\)
\(13\)	−3.59591	−	0.263474i	−0.997326	−	0.0730746i
							\(17\)	−3.29258	−	5.97777i	−0.798568	−	1.44982i	−0.890951	−	0.454099i	\(-0.849961\pi\)
0.0923828	−	0.995724i	\(-0.470552\pi\)
\(19\)	1.76122	−	0.471917i	0.404051	−	0.108265i	−0.0510692	−	0.998695i	\(-0.516263\pi\)
							0.455120	+	0.890430i	\(0.349596\pi\)
\(23\)	0.198025	−	0.739040i	0.0412911	−	0.154101i	−0.942202	−	0.335045i	\(-0.891249\pi\)
							0.983493	+	0.180944i	\(0.0579153\pi\)
\(29\)	8.70393	+	4.13006i	1.61628	+	0.766934i	0.999791	−	0.0204309i	\(-0.00650381\pi\)
							0.616487	+	0.787365i	\(0.288555\pi\)
\(31\)	6.09210	+	4.77286i	1.09417	+	0.857230i	0.990518	−	0.137383i	\(-0.0438691\pi\)
							0.103656	+	0.994613i	\(0.466946\pi\)
\(37\)	3.65498	+	4.86389i	0.600875	+	0.799619i	0.992724	−	0.120410i	\(-0.0384211\pi\)
							−0.391850	+	0.920029i	\(0.628165\pi\)
\(41\)	−1.00749	−	0.929433i	−0.157343	−	0.145153i	0.595494	−	0.803360i	\(-0.296956\pi\)
							−0.752837	+	0.658207i	\(0.771315\pi\)
\(43\)	0.167782	+	1.18231i	0.0255866	+	0.180301i	0.998761	−	0.0497682i	\(-0.0158483\pi\)
							−0.973174	+	0.230069i	\(0.926105\pi\)
\(47\)	−9.61761	+	3.64748i	−1.40287	+	0.532040i	−0.935911	−	0.352238i	\(-0.885421\pi\)
							−0.466962	+	0.884277i	\(0.654652\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.23	yes	4272
5.3	odd	4		845.2.bi.a.223.67	yes	4272
169.72	odd	156		845.2.bi.a.72.67	✓	4272
845.748	even	156	inner	845.2.bn.a.748.23	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.67	✓	4272	169.72	odd	156
845.2.bi.a.223.67	yes	4272	5.3	odd	4
845.2.bn.a.392.23	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.23	yes	4272	845.748	even	156	inner