Embedded newform 845.2.bn.a.392.34

• Label: 845.2.bn.a.392.34
• 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			392.34
Character	\(\chi\)	\(=\)	845.392
Dual form			845.2.bn.a.748.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.334813 + 0.705603i) q^{2} +(-2.30452 - 2.12598i) q^{3} +(0.879115 + 1.07672i) q^{4} +(-2.22387 - 0.233270i) q^{5} +(2.27168 - 0.914272i) q^{6} +(-0.816504 - 0.236503i) q^{7} +(-2.57070 + 0.633622i) q^{8} +(0.549635 + 6.80846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4272 q - 48 q^{2} - 50 q^{3} + 174 q^{4} - 46 q^{5} - 96 q^{6} - 46 q^{7} - 64 q^{8} + 12 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.334813	+	0.705603i	−0.236748	+	0.498936i	−0.986699	−	0.162557i	\(-0.948026\pi\)
							0.749951	+	0.661494i	\(0.230077\pi\)
\(3\)	−2.30452	−	2.12598i	−1.33052	−	1.22743i	−0.953560	−	0.301203i	\(-0.902612\pi\)
							−0.376956	−	0.926231i	\(-0.623029\pi\)
\(5\)	−2.22387	−	0.233270i	−0.994544	−	0.104322i
							\(7\)	−0.816504	−	0.236503i	−0.308610	−	0.0893898i	0.120299	−	0.992738i	\(-0.461615\pi\)
−0.428909	+	0.903348i	\(0.641102\pi\)
\(11\)	−0.783481	−	0.921088i	−0.236228	−	0.277718i	0.630834	−	0.775918i	\(-0.282713\pi\)
							−0.867063	+	0.498199i	\(0.833995\pi\)
\(13\)	0.957258	+	3.47616i	0.265495	+	0.964112i
							\(17\)	−0.560528	−	1.01765i	−0.135948	−	0.246817i	0.799207	−	0.601056i	\(-0.205253\pi\)
−0.935155	+	0.354238i	\(0.884740\pi\)
\(19\)	0.675405	−	0.180974i	0.154949	−	0.0415183i	−0.180511	−	0.983573i	\(-0.557775\pi\)
							0.335459	+	0.942055i	\(0.391108\pi\)
\(23\)	0.920365	−	3.43485i	0.191909	−	0.716215i	−0.801136	−	0.598482i	\(-0.795771\pi\)
							0.993045	−	0.117733i	\(-0.0375627\pi\)
\(29\)	−4.98424	−	2.36505i	−0.925551	−	0.439179i	−0.0945938	−	0.995516i	\(-0.530155\pi\)
							−0.830957	+	0.556337i	\(0.812207\pi\)
\(31\)	6.20117	+	4.85831i	1.11376	+	0.872578i	0.992765	−	0.120075i	\(-0.0383135\pi\)
							0.120998	+	0.992653i	\(0.461390\pi\)
\(37\)	−2.44815	−	3.25789i	−0.402473	−	0.535594i	0.552019	−	0.833832i	\(-0.313858\pi\)
							−0.954492	+	0.298237i	\(0.903601\pi\)
\(41\)	−0.675883	−	0.623519i	−0.105555	−	0.0973773i	0.623533	−	0.781797i	\(-0.285697\pi\)
							−0.729088	+	0.684420i	\(0.760056\pi\)
\(43\)	−1.01971	−	7.18557i	−0.155504	−	1.09579i	−0.902416	−	0.430866i	\(-0.858208\pi\)
							0.746912	−	0.664923i	\(-0.231536\pi\)
\(47\)	8.44041	−	3.20103i	1.23116	−	0.466918i	0.348669	−	0.937246i	\(-0.386634\pi\)
							0.882492	+	0.470328i	\(0.155864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bn.a.392.34	yes	4272
5.3	odd	4		845.2.bi.a.223.56	yes	4272
169.72	odd	156		845.2.bi.a.72.56	✓	4272
845.748	even	156	inner	845.2.bn.a.748.34	yes	4272

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bi.a.72.56	✓	4272	169.72	odd	156
845.2.bi.a.223.56	yes	4272	5.3	odd	4
845.2.bn.a.392.34	yes	4272	1.1	even	1	trivial
845.2.bn.a.748.34	yes	4272	845.748	even	156	inner