Embedded newform 845.2.bg.a.381.8

• Label: 845.2.bg.a.381.8
• Level: $845$
• Weight: $2$
• Character: 845.381
• 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			381.8
Character	\(\chi\)	\(=\)	845.381
Dual form			845.2.bg.a.621.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81669 + 1.48328i) q^{2} +(2.31724 - 0.187067i) q^{3} +(0.700192 - 3.42977i) q^{4} +(0.992709 + 0.120537i) q^{5} +(-3.93224 + 3.77697i) q^{6} +(-1.01235 + 1.60091i) q^{7} +(1.63544 + 3.11607i) q^{8} +(2.37346 - 0.385725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(171\)	\(677\)
\(\chi(n)\)	\(e\left(\frac{61}{78}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81669	+	1.48328i	−1.28460	+	1.04884i	−0.288786	+	0.957394i	\(0.593252\pi\)
							−0.995810	+	0.0914470i	\(0.970851\pi\)
\(3\)	2.31724	−	0.187067i	1.33786	−	0.108003i	0.609275	−	0.792959i	\(-0.291461\pi\)
							0.728585	+	0.684956i	\(0.240178\pi\)
\(5\)	0.992709	+	0.120537i	0.443953	+	0.0539056i
							\(7\)	−1.01235	+	1.60091i	−0.382634	+	0.605087i	−0.980502	−	0.196507i	\(-0.937040\pi\)
0.597868	+	0.801594i	\(0.296015\pi\)
\(11\)	−0.802936	+	4.94067i	−0.242094	+	1.48967i	0.526634	+	0.850092i	\(0.323454\pi\)
							−0.768729	+	0.639575i	\(0.779110\pi\)
\(13\)	1.52988	+	3.26488i	0.424313	+	0.905515i
							\(17\)	−0.255210	−	0.161385i	−0.0618976	−	0.0391417i	0.503159	−	0.864194i	\(-0.332171\pi\)
−0.565056	+	0.825052i	\(0.691146\pi\)
\(19\)	−2.60166	+	1.50207i	−0.596861	+	0.344598i	−0.767806	−	0.640683i	\(-0.778651\pi\)
							0.170945	+	0.985281i	\(0.445318\pi\)
\(23\)	3.17393	−	5.49741i	0.661810	−	1.14629i	−0.318329	−	0.947980i	\(-0.603122\pi\)
							0.980140	−	0.198309i	\(-0.0635449\pi\)
\(29\)	4.15866	+	5.09343i	0.772244	+	0.945827i	0.999608	−	0.0279829i	\(-0.00890838\pi\)
							−0.227365	+	0.973810i	\(0.573011\pi\)
\(31\)	−1.77546	−	7.20334i	−0.318883	−	1.29376i	−0.883510	−	0.468412i	\(-0.844826\pi\)
							0.564627	−	0.825346i	\(-0.309020\pi\)
\(37\)	−9.84499	−	2.85163i	−1.61851	−	0.468806i	−0.659385	−	0.751805i	\(-0.729183\pi\)
							−0.959120	+	0.282999i	\(0.908671\pi\)
\(41\)	0.182813	+	2.26454i	0.0285506	+	0.353662i	0.995222	+	0.0976409i	\(0.0311296\pi\)
							−0.966671	+	0.256021i	\(0.917588\pi\)
\(43\)	−0.792581	−	2.73631i	−0.120868	−	0.417283i	0.877150	−	0.480216i	\(-0.159442\pi\)
							−0.998018	+	0.0629330i	\(0.979955\pi\)
\(47\)	6.85935	+	7.74261i	1.00054	+	1.12938i	0.991590	+	0.129416i	\(0.0413102\pi\)
							0.00894913	+	0.999960i	\(0.497151\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.381.8	✓	1440
169.114	even	78	inner	845.2.bg.a.621.8	yes	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.8	✓	1440	1.1	even	1	trivial
845.2.bg.a.621.8	yes	1440	169.114	even	78	inner