Embedded newform 864.2.bu.a.13.11

• Label: 864.2.bu.a.13.11
• Level: $864$
• Weight: $2$
• Character: 864.13
• Analytic conductor: $6.899$
• Analytic rank: $0$
• Dimension: $3408$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.bu (of order \(72\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.89907473464\)
Analytic rank:	\(0\)
Dimension:	\(3408\)
Relative dimension:	\(142\) over \(\Q(\zeta_{72})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{72}]$

Embedding invariants

Embedding label			13.11
Character	\(\chi\)	\(=\)	864.13
Dual form			864.2.bu.a.133.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.37712 - 0.321786i) q^{2} +(0.390709 + 1.68741i) q^{3} +(1.79291 + 0.886276i) q^{4} +(3.68959 + 1.16332i) q^{5} +(0.00493340 - 2.44948i) q^{6} +(-0.687024 + 0.320365i) q^{7} +(-2.18385 - 1.79744i) q^{8} +(-2.69469 + 1.31857i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3408 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 12 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(353\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{4}{9}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.37712	−	0.321786i	−0.973769	−	0.227537i
							\(3\)	0.390709	+	1.68741i	0.225576	+	0.974226i
\(5\)	3.68959	+	1.16332i	1.65003	+	0.520254i								0.975875	−	0.218331i	\(-0.0700613\pi\)
							0.674160	+	0.738585i	\(0.264506\pi\)
\(7\)	−0.687024	+	0.320365i	−0.259671	+	0.121086i	−0.548095	−	0.836416i	\(-0.684647\pi\)
							0.288424	+	0.957503i	\(0.406869\pi\)
\(11\)	1.72473	+	0.897838i	0.520026	+	0.270708i	0.701478	−	0.712691i	\(-0.252524\pi\)
							−0.181452	+	0.983400i	\(0.558080\pi\)
\(13\)	1.78622	−	2.80381i	0.495409	−	0.777636i	−0.500713	−	0.865613i	\(-0.666929\pi\)
							0.996123	+	0.0879766i	\(0.0280401\pi\)
\(17\)	−3.33001	−	1.92258i	−0.807646	−	0.466295i	0.0384916	−	0.999259i	\(-0.487745\pi\)
							−0.846138	+	0.532964i	\(0.821078\pi\)
\(19\)	0.967745	+	7.35075i	0.222016	+	1.68638i	0.632539	+	0.774528i	\(0.282013\pi\)
							−0.410523	+	0.911850i	\(0.634654\pi\)
\(23\)	8.09302	+	3.77384i	1.68751	+	0.786899i	0.998040	+	0.0625867i	\(0.0199350\pi\)
							0.689472	+	0.724313i	\(0.257843\pi\)
\(29\)	−8.63809	−	1.91502i	−1.60405	−	0.355610i	−0.678610	−	0.734499i	\(-0.737417\pi\)
							−0.925442	+	0.378889i	\(0.876306\pi\)
\(31\)	5.47591	−	1.99307i	0.983502	−	0.357966i	0.200301	−	0.979734i	\(-0.435808\pi\)
							0.783201	+	0.621769i	\(0.213586\pi\)
\(37\)	−0.844049	+	6.41119i	−0.138761	+	1.05399i	0.770404	+	0.637556i	\(0.220055\pi\)
							−0.909165	+	0.416437i	\(0.863279\pi\)
\(41\)	3.08206	−	2.15808i	0.481338	−	0.337036i	−0.307594	−	0.951518i	\(-0.599524\pi\)
							0.788931	+	0.614482i	\(0.210635\pi\)
\(43\)	−3.37872	−	1.75885i	−0.515249	−	0.268222i	0.184219	−	0.982885i	\(-0.441024\pi\)
							−0.699469	+	0.714663i	\(0.746580\pi\)
\(47\)	2.02510	−	5.56390i	0.295390	−	0.811579i	−0.699864	−	0.714276i	\(-0.746756\pi\)
							0.995255	−	0.0973029i	\(-0.0310216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.bu.a.13.11	✓	3408
27.25	even	9	inner	864.2.bu.a.781.117	yes	3408
32.5	even	8	inner	864.2.bu.a.229.117	yes	3408
864.133	even	72	inner	864.2.bu.a.133.11	yes	3408

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.2.bu.a.13.11	✓	3408	1.1	even	1	trivial
864.2.bu.a.133.11	yes	3408	864.133	even	72	inner
864.2.bu.a.229.117	yes	3408	32.5	even	8	inner
864.2.bu.a.781.117	yes	3408	27.25	even	9	inner