Embedded newform 648.2.bd.a.133.26

• Label: 648.2.bd.a.133.26
• Level: $648$
• Weight: $2$
• Character: 648.133
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bd (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1908\)
Relative dimension:	\(106\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			133.26
Character	\(\chi\)	\(=\)	648.133
Dual form			648.2.bd.a.229.26

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02131 - 0.978229i) q^{2} +(-0.991055 + 1.42050i) q^{3} +(0.0861370 + 1.99814i) q^{4} +(0.499376 - 0.471137i) q^{5} +(2.40174 - 0.481285i) q^{6} +(-3.08454 + 0.360531i) q^{7} +(1.86667 - 2.12498i) q^{8} +(-1.03562 - 2.81558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02131	−	0.978229i	−0.722173	−	0.691712i
							\(3\)	−0.991055	+	1.42050i	−0.572186	+	0.820124i
\(5\)	0.499376	−	0.471137i	0.223328	−	0.210699i								−0.566443	−	0.824101i	\(-0.691681\pi\)
							0.789771	+	0.613402i	\(0.210199\pi\)
\(7\)	−3.08454	+	0.360531i	−1.16585	+	0.136268i	−0.676919	−	0.736057i	\(-0.736685\pi\)
							−0.488927	+	0.872325i	\(0.662611\pi\)
\(11\)	1.31829	−	0.394670i	0.397479	−	0.118997i	−0.0818242	−	0.996647i	\(-0.526075\pi\)
							0.479303	+	0.877649i	\(0.340889\pi\)
\(13\)	4.56174	−	0.265691i	1.26520	−	0.0736894i	0.587618	−	0.809139i	\(-0.300066\pi\)
							0.677580	+	0.735449i	\(0.263029\pi\)
\(17\)	4.36263	+	1.58787i	1.05809	+	0.385115i	0.811712	−	0.584058i	\(-0.198536\pi\)
							0.246382	+	0.969173i	\(0.420758\pi\)
\(19\)	−1.77363	−	4.87300i	−0.406898	−	1.11794i	−0.958812	−	0.284042i	\(-0.908325\pi\)
							0.551914	−	0.833901i	\(-0.313898\pi\)
\(23\)	−1.14756	−	0.134130i	−0.239282	−	0.0279681i	−0.00439351	−	0.999990i	\(-0.501399\pi\)
							−0.234889	+	0.972022i	\(0.575473\pi\)
\(29\)	4.54696	+	9.05374i	0.844349	+	1.68124i	0.725390	+	0.688338i	\(0.241660\pi\)
							0.118960	+	0.992899i	\(0.462044\pi\)
\(31\)	−6.18976	+	8.31428i	−1.11171	+	1.49329i	−0.262730	+	0.964869i	\(0.584623\pi\)
							−0.848983	+	0.528420i	\(0.822785\pi\)
\(37\)	0.0755451	+	0.0900311i	0.0124195	+	0.0148010i	0.772218	−	0.635357i	\(-0.219147\pi\)
							−0.759799	+	0.650158i	\(0.774703\pi\)
\(41\)	−4.06663	+	2.67467i	−0.635101	+	0.417713i	−0.825821	−	0.563932i	\(-0.809288\pi\)
							0.190720	+	0.981644i	\(0.438918\pi\)
\(43\)	0.380033	−	1.60348i	0.0579544	−	0.244529i	−0.936356	−	0.351053i	\(-0.885824\pi\)
							0.994310	+	0.106524i	\(0.0339720\pi\)
\(47\)	3.75261	+	5.04063i	0.547375	+	0.735252i	0.986606	−	0.163122i	\(-0.0521564\pi\)
							−0.439231	+	0.898374i	\(0.644749\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.133.26	yes	1908
8.5	even	2	inner	648.2.bd.a.133.15	✓	1908
81.67	even	27	inner	648.2.bd.a.229.15	yes	1908
648.229	even	54	inner	648.2.bd.a.229.26	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.15	✓	1908	8.5	even	2	inner
648.2.bd.a.133.26	yes	1908	1.1	even	1	trivial
648.2.bd.a.229.15	yes	1908	81.67	even	27	inner
648.2.bd.a.229.26	yes	1908	648.229	even	54	inner