Embedded newform 845.2.bg.a.621.27

• Label: 845.2.bg.a.621.27
• 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			621.27
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.27

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.380707 - 0.310837i) q^{2} +(1.60690 + 0.129723i) q^{3} +(-0.351734 - 1.72291i) q^{4} +(-0.992709 + 0.120537i) q^{5} +(-0.571436 - 0.548872i) q^{6} +(0.406994 + 0.643610i) q^{7} +(-0.858445 + 1.63563i) q^{8} +(-0.395837 - 0.0643297i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.380707	−	0.310837i	−0.269200	−	0.219795i	0.488224	−	0.872718i	\(-0.337645\pi\)
							−0.757424	+	0.652923i	\(0.773542\pi\)
\(3\)	1.60690	+	0.129723i	0.927747	+	0.0748955i	0.535095	−	0.844792i	\(-0.320276\pi\)
							0.392652	+	0.919687i	\(0.371558\pi\)
\(5\)	−0.992709	+	0.120537i	−0.443953	+	0.0539056i
							\(7\)	0.406994	+	0.643610i	0.153829	+	0.243262i	0.913307	−	0.407272i	\(-0.133520\pi\)
−0.759478	+	0.650533i	\(0.774545\pi\)
\(11\)	−0.635613	−	3.91108i	−0.191644	−	1.17924i	−0.887415	−	0.460971i	\(-0.847501\pi\)
							0.695771	−	0.718264i	\(-0.255063\pi\)
\(13\)	−2.89483	−	2.14940i	−0.802883	−	0.596137i
							\(17\)	−2.58304	+	1.63342i	−0.626479	+	0.396162i	−0.809645	−	0.586921i	\(-0.800340\pi\)
0.183165	+	0.983082i	\(0.441366\pi\)
\(19\)	−1.59150	−	0.918853i	−0.365115	−	0.210799i	0.306207	−	0.951965i	\(-0.400940\pi\)
							−0.671322	+	0.741166i	\(0.734273\pi\)
\(23\)	0.418354	+	0.724611i	0.0872329	+	0.151092i	0.906341	−	0.422548i	\(-0.138864\pi\)
							−0.819108	+	0.573640i	\(0.805531\pi\)
\(29\)	1.31552	−	1.61122i	0.244286	−	0.299196i	−0.637879	−	0.770137i	\(-0.720188\pi\)
							0.882165	+	0.470941i	\(0.156085\pi\)
\(31\)	2.08073	−	8.44184i	0.373709	−	1.51620i	−0.419786	−	0.907623i	\(-0.637895\pi\)
							0.793496	−	0.608576i	\(-0.208259\pi\)
\(37\)	−5.08883	+	1.47400i	−0.836599	+	0.242324i	−0.668748	−	0.743489i	\(-0.733170\pi\)
							−0.167850	+	0.985813i	\(0.553683\pi\)
\(41\)	0.377837	−	4.68035i	0.0590082	−	0.730948i	−0.897833	−	0.440336i	\(-0.854859\pi\)
							0.956841	−	0.290612i	\(-0.0938588\pi\)
\(43\)	−0.785877	+	2.71316i	−0.119845	+	0.413753i	−0.997898	−	0.0648110i	\(-0.979356\pi\)
							0.878052	+	0.478565i	\(0.158843\pi\)
\(47\)	−1.94748	+	2.19825i	−0.284069	+	0.320648i	−0.873174	−	0.487409i	\(-0.837942\pi\)
							0.589105	+	0.808057i	\(0.299481\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.27	yes	1440
169.43	even	78	inner	845.2.bg.a.381.27	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.27	✓	1440	169.43	even	78	inner
845.2.bg.a.621.27	yes	1440	1.1	even	1	trivial