Embedded newform 845.2.bg.a.621.19

• Label: 845.2.bg.a.621.19
• Level: $845$
• Weight: $2$
• Character: 845.621
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $1440$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bg (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(1440\)
Relative dimension:	\(60\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

Embedding invariants

Embedding label			621.19
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.992882 - 0.810663i) q^{2} +(0.0564152 + 0.00455430i) q^{3} +(-0.0714114 - 0.349796i) q^{4} +(0.992709 - 0.120537i) q^{5} +(-0.0523216 - 0.0502556i) q^{6} +(1.35693 + 2.14582i) q^{7} +(-1.40402 + 2.67514i) q^{8} +(-2.95799 - 0.480720i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{17}{78}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.992882	−	0.810663i	−0.702073	−	0.573225i	0.212737	−	0.977110i	\(-0.431762\pi\)
							−0.914810	+	0.403884i	\(0.867660\pi\)
\(3\)	0.0564152	+	0.00455430i	0.0325713	+	0.00262943i	0.0967415	−	0.995310i	\(-0.469158\pi\)
							−0.0641702	+	0.997939i	\(0.520440\pi\)
\(5\)	0.992709	−	0.120537i	0.443953	−	0.0539056i
							\(7\)	1.35693	+	2.14582i	0.512873	+	0.811044i	0.997988	−	0.0634067i	\(-0.0201965\pi\)
−0.485115	+	0.874451i	\(0.661222\pi\)
\(11\)	−0.236027	−	1.45233i	−0.0711649	−	0.437895i	−0.998043	−	0.0625325i	\(-0.980082\pi\)
							0.926878	−	0.375363i	\(-0.122482\pi\)
\(13\)	3.11182	+	1.82115i	0.863063	+	0.505097i
							\(17\)	−5.74559	+	3.63329i	−1.39351	+	0.881202i	−0.999285	−	0.0378119i	\(-0.987961\pi\)
−0.394225	+	0.919014i	\(0.628987\pi\)
\(19\)	4.17914	+	2.41283i	0.958760	+	0.553540i	0.895791	−	0.444475i	\(-0.146610\pi\)
							0.0629688	+	0.998015i	\(0.479943\pi\)
\(23\)	1.21334	+	2.10156i	0.252998	+	0.438205i	0.964350	−	0.264631i	\(-0.0852501\pi\)
							−0.711352	+	0.702836i	\(0.751917\pi\)
\(29\)	−3.96354	+	4.85445i	−0.736010	+	0.901449i	−0.997974	−	0.0636287i	\(-0.979733\pi\)
							0.261963	+	0.965078i	\(0.415630\pi\)
\(31\)	−1.69807	+	6.88936i	−0.304983	+	1.23737i	0.595813	+	0.803123i	\(0.296830\pi\)
							−0.900796	+	0.434242i	\(0.857016\pi\)
\(37\)	−5.60646	+	1.62393i	−0.921696	+	0.266973i	−0.704951	−	0.709256i	\(-0.749031\pi\)
							−0.216745	+	0.976228i	\(0.569544\pi\)
\(41\)	0.414326	−	5.13234i	0.0647068	−	0.801537i	−0.880285	−	0.474444i	\(-0.842649\pi\)
							0.944992	−	0.327093i	\(-0.106069\pi\)
\(43\)	0.222130	−	0.766882i	0.0338745	−	0.116948i	−0.941799	−	0.336176i	\(-0.890866\pi\)
							0.975674	+	0.219227i	\(0.0703536\pi\)
\(47\)	2.72124	−	3.07165i	0.396934	−	0.448046i	−0.515733	−	0.856749i	\(-0.672480\pi\)
							0.912667	+	0.408703i	\(0.134019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.19	yes	1440
169.43	even	78	inner	845.2.bg.a.381.19	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.19	✓	1440	169.43	even	78	inner
845.2.bg.a.621.19	yes	1440	1.1	even	1	trivial