Embedded newform 845.2.bg.a.621.23

• Label: 845.2.bg.a.621.23
• 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			621.23
Character	\(\chi\)	\(=\)	845.621
Dual form			845.2.bg.a.381.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.565165 - 0.461443i) q^{2} +(-1.03665 - 0.0836872i) q^{3} +(-0.293570 - 1.43800i) q^{4} +(-0.992709 + 0.120537i) q^{5} +(0.547262 + 0.525652i) q^{6} +(-1.93350 - 3.05759i) q^{7} +(-1.17578 + 2.24026i) q^{8} +(-1.89351 - 0.307725i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1440 q - 2 q^{3} - 58 q^{4} + 18 q^{6} + 6 q^{7} + 60 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.565165	−	0.461443i	−0.399632	−	0.326289i	0.411225	−	0.911534i	\(-0.365101\pi\)
							−0.810857	+	0.585245i	\(0.800998\pi\)
\(3\)	−1.03665	−	0.0836872i	−0.598511	−	0.0483168i	−0.222501	−	0.974932i	\(-0.571422\pi\)
							−0.376010	+	0.926616i	\(0.622704\pi\)
\(5\)	−0.992709	+	0.120537i	−0.443953	+	0.0539056i
							\(7\)	−1.93350	−	3.05759i	−0.730795	−	1.15566i	−0.982211	−	0.187780i	\(-0.939871\pi\)
0.251416	−	0.967879i	\(-0.419104\pi\)
\(11\)	0.284989	+	1.75361i	0.0859273	+	0.528732i	0.993952	+	0.109815i	\(0.0350259\pi\)
							−0.908025	+	0.418917i	\(0.862410\pi\)
\(13\)	−1.77463	+	3.13858i	−0.492194	+	0.870486i
							\(17\)	−0.841460	+	0.532107i	−0.204084	+	0.129055i	−0.632598	−	0.774480i	\(-0.718012\pi\)
0.428514	+	0.903535i	\(0.359037\pi\)
\(19\)	5.07798	+	2.93177i	1.16497	+	0.672595i	0.952490	−	0.304571i	\(-0.0985131\pi\)
							0.212479	+	0.977166i	\(0.431846\pi\)
\(23\)	−4.70648	−	8.15187i	−0.981370	−	1.69978i	−0.657074	−	0.753826i	\(-0.728206\pi\)
							−0.324295	−	0.945956i	\(-0.605127\pi\)
\(29\)	5.04725	−	6.18176i	0.937251	−	1.14792i	−0.0515484	−	0.998670i	\(-0.516416\pi\)
							0.988799	−	0.149253i	\(-0.0476869\pi\)
\(31\)	−1.73361	+	7.03352i	−0.311365	+	1.26326i	0.581727	+	0.813384i	\(0.302377\pi\)
							−0.893092	+	0.449874i	\(0.851469\pi\)
\(37\)	5.75374	−	1.66659i	0.945910	−	0.273986i	0.230802	−	0.973001i	\(-0.425865\pi\)
							0.715107	+	0.699015i	\(0.246378\pi\)
\(41\)	0.718695	−	8.90264i	0.112241	−	1.39036i	−0.658198	−	0.752845i	\(-0.728681\pi\)
							0.770439	−	0.637513i	\(-0.220037\pi\)
\(43\)	−1.65447	+	5.71188i	−0.252304	+	0.871053i	0.729951	+	0.683500i	\(0.239543\pi\)
							−0.982254	+	0.187553i	\(0.939944\pi\)
\(47\)	−2.96974	+	3.35214i	−0.433180	+	0.488960i	−0.924049	−	0.382274i	\(-0.875141\pi\)
							0.490869	+	0.871234i	\(0.336680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.bg.a.621.23	yes	1440
169.43	even	78	inner	845.2.bg.a.381.23	✓	1440

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.bg.a.381.23	✓	1440	169.43	even	78	inner
845.2.bg.a.621.23	yes	1440	1.1	even	1	trivial