Embedded newform 845.2.bn.a.392.66

• Label: 845.2.bn.a.392.66
• 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			392.66
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.66

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.658703 - 1.38819i) q^{2} +(-1.90949 - 1.76155i) q^{3} +(-0.228284 - 0.279597i) q^{4} +(-2.23425 + 0.0902724i) q^{5} +(-3.70316 + 1.49039i) q^{6} +(3.79313 + 1.09869i) q^{7} +(2.44528 - 0.602707i) q^{8} +(0.301688 + 3.73708i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.658703	−	1.38819i	0.465773	−	0.981597i	−0.525625	−	0.850717i	\(-0.676168\pi\)
							0.991398	−	0.130880i	\(-0.0417803\pi\)
\(3\)	−1.90949	−	1.76155i	−1.10245	−	1.01703i	−0.999751	−	0.0223214i	\(-0.992894\pi\)
							−0.102696	−	0.994713i	\(-0.532747\pi\)
\(5\)	−2.23425	+	0.0902724i	−0.999185	+	0.0403710i
							\(7\)	3.79313	+	1.09869i	1.43367	+	0.415267i	0.902019	−	0.431697i	\(-0.142085\pi\)
0.531650	+	0.846964i	\(0.321572\pi\)
\(11\)	1.51930	+	1.78614i	0.458085	+	0.538541i	0.941572	−	0.336810i	\(-0.109348\pi\)
							−0.483488	+	0.875351i	\(0.660630\pi\)
\(13\)	−0.214631	+	3.59916i	−0.0595280	+	0.998227i
							\(17\)	2.92190	+	5.30480i	0.708666	+	1.28660i	0.948804	+	0.315864i	\(0.102294\pi\)
−0.240139	+	0.970739i	\(0.577193\pi\)
\(19\)	3.29079	−	0.881764i	0.754958	−	0.202290i	0.139242	−	0.990258i	\(-0.455533\pi\)
							0.615716	+	0.787968i	\(0.288867\pi\)
\(23\)	−0.740123	+	2.76218i	−0.154326	+	0.575954i	0.844836	+	0.535026i	\(0.179698\pi\)
							−0.999162	+	0.0409280i	\(0.986969\pi\)
\(29\)	1.60550	+	0.761821i	0.298135	+	0.141467i	0.571884	−	0.820334i	\(-0.306213\pi\)
							−0.273750	+	0.961801i	\(0.588264\pi\)
\(31\)	−6.93496	−	5.43320i	−1.24556	−	0.975831i	−0.999979	−	0.00645379i	\(-0.997946\pi\)
							−0.245577	−	0.969377i	\(-0.578977\pi\)
\(37\)	0.303882	+	0.404394i	0.0499579	+	0.0664819i	0.823730	−	0.566982i	\(-0.191889\pi\)
							−0.773772	+	0.633464i	\(0.781633\pi\)
\(41\)	5.80704	+	5.35713i	0.906907	+	0.836644i	0.987095	−	0.160136i	\(-0.0511932\pi\)
							−0.0801884	+	0.996780i	\(0.525552\pi\)
\(43\)	−1.70145	−	11.9896i	−0.259468	−	1.82840i	−0.509865	−	0.860254i	\(-0.670305\pi\)
							0.250397	−	0.968143i	\(-0.419439\pi\)
\(47\)	−4.44088	+	1.68420i	−0.647769	+	0.245666i	−0.656535	−	0.754295i	\(-0.727979\pi\)
							0.00876643	+	0.999962i	\(0.497210\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.66	yes	4272
5.3	odd	4		845.2.bi.a.223.24	yes	4272
169.72	odd	156		845.2.bi.a.72.24	✓	4272
845.748	even	156	inner	845.2.bn.a.748.66	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.24	✓	4272	169.72	odd	156
845.2.bi.a.223.24	yes	4272	5.3	odd	4
845.2.bn.a.392.66	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.66	yes	4272	845.748	even	156	inner