Embedded newform 845.2.bg.a.621.17

• Label: 845.2.bg.a.621.17
• 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			621.17
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15912 - 0.946389i) q^{2} +(2.97310 + 0.240013i) q^{3} +(0.0478466 + 0.234368i) q^{4} +(-0.992709 + 0.120537i) q^{5} +(-3.21902 - 3.09191i) q^{6} +(1.37836 + 2.17970i) q^{7} +(-1.22448 + 2.33304i) q^{8} +(5.82056 + 0.945933i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15912	−	0.946389i	−0.819619	−	0.669198i	0.126875	−	0.991919i	\(-0.459505\pi\)
							−0.946494	+	0.322721i	\(0.895403\pi\)
\(3\)	2.97310	+	0.240013i	1.71652	+	0.138572i	0.899178	−	0.437583i	\(-0.144165\pi\)
							0.817341	+	0.576154i	\(0.195447\pi\)
\(5\)	−0.992709	+	0.120537i	−0.443953	+	0.0539056i
							\(7\)	1.37836	+	2.17970i	0.520971	+	0.823850i	0.998499	−	0.0547668i	\(-0.0174415\pi\)
−0.477528	+	0.878616i	\(0.658467\pi\)
\(11\)	0.340897	+	2.09763i	0.102784	+	0.632458i	0.986415	+	0.164273i	\(0.0525278\pi\)
							−0.883631	+	0.468185i	\(0.844908\pi\)
\(13\)	2.80712	−	2.26275i	0.778556	−	0.627575i
							\(17\)	−5.15081	+	3.25718i	−1.24926	+	0.789981i	−0.984678	−	0.174384i	\(-0.944207\pi\)
−0.264577	+	0.964365i	\(0.585232\pi\)
\(19\)	2.85651	+	1.64921i	0.655329	+	0.378355i	0.790495	−	0.612468i	\(-0.209823\pi\)
							−0.135166	+	0.990823i	\(0.543157\pi\)
\(23\)	0.642722	+	1.11323i	0.134017	+	0.232124i	0.925221	−	0.379428i	\(-0.123879\pi\)
							−0.791205	+	0.611551i	\(0.790546\pi\)
\(29\)	2.60296	−	3.18805i	0.483358	−	0.592006i	−0.474139	−	0.880450i	\(-0.657241\pi\)
							0.957497	+	0.288444i	\(0.0931380\pi\)
\(31\)	−1.82225	+	7.39318i	−0.327287	+	1.32785i	0.544826	+	0.838549i	\(0.316596\pi\)
							−0.872113	+	0.489305i	\(0.837251\pi\)
\(37\)	3.37220	−	0.976769i	0.554386	−	0.160580i	0.0108127	−	0.999942i	\(-0.496558\pi\)
							0.543573	+	0.839362i	\(0.317071\pi\)
\(41\)	−0.0981344	+	1.21561i	−0.0153260	+	0.189847i	0.984564	+	0.175024i	\(0.0560004\pi\)
							−0.999890	+	0.0148225i	\(0.995282\pi\)
\(43\)	3.05393	−	10.5434i	0.465720	−	1.60785i	−0.292782	−	0.956179i	\(-0.594581\pi\)
							0.758501	−	0.651671i	\(-0.225932\pi\)
\(47\)	−0.0219646	+	0.0247929i	−0.00320386	+	0.00361641i	−0.750110	−	0.661313i	\(-0.770000\pi\)
							0.746907	+	0.664929i	\(0.231538\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.17	yes	1440
169.43	even	78	inner	845.2.bg.a.381.17	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.17	✓	1440	169.43	even	78	inner
845.2.bg.a.621.17	yes	1440	1.1	even	1	trivial