Embedded newform 648.2.bb.b.131.48

• Label: 648.2.bb.b.131.48
• Level: $648$
• Weight: $2$
• Character: 648.131
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1872$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1872\)
Relative dimension:	\(104\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			131.48
Character	\(\chi\)	\(=\)	648.131
Dual form			648.2.bb.b.371.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.269596 + 1.38828i) q^{2} +(-1.12977 - 1.31287i) q^{3} +(-1.85464 - 0.748549i) q^{4} +(-0.173218 + 2.97403i) q^{5} +(2.12721 - 1.21449i) q^{6} +(-0.435924 - 1.83931i) q^{7} +(1.53920 - 2.37295i) q^{8} +(-0.447242 + 2.96648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{47}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.269596	+	1.38828i	−0.190633	+	0.981661i
							\(3\)	−1.12977	−	1.31287i	−0.652273	−	0.757984i
\(5\)	−0.173218	+	2.97403i	−0.0774653	+	1.33003i								0.705831	+	0.708380i	\(0.250574\pi\)
							−0.783296	+	0.621648i	\(0.786463\pi\)
\(7\)	−0.435924	−	1.83931i	−0.164764	−	0.695193i	−0.991223	−	0.132197i	\(-0.957797\pi\)
							0.826460	−	0.562996i	\(-0.190351\pi\)
\(11\)	0.120089	+	0.182586i	0.0362081	+	0.0550518i	0.853138	−	0.521685i	\(-0.174696\pi\)
							−0.816930	+	0.576737i	\(0.804326\pi\)
\(13\)	−0.773185	−	6.61502i	−0.214443	−	1.83468i	−0.488027	−	0.872828i	\(-0.662283\pi\)
							0.273584	−	0.961848i	\(-0.411791\pi\)
\(17\)	−4.15387	+	4.95039i	−1.00746	+	1.20065i	−0.0278766	+	0.999611i	\(0.508875\pi\)
							−0.979584	+	0.201034i	\(0.935570\pi\)
\(19\)	2.45070	−	2.05638i	0.562228	−	0.471766i	−0.316828	−	0.948483i	\(-0.602618\pi\)
							0.879057	+	0.476717i	\(0.158173\pi\)
\(23\)	0.420938	+	0.0997641i	0.0877716	+	0.0208023i	0.274267	−	0.961654i	\(-0.411565\pi\)
							−0.186495	+	0.982456i	\(0.559713\pi\)
\(29\)	4.91082	−	6.59638i	0.911916	−	1.22492i	−0.0620153	−	0.998075i	\(-0.519753\pi\)
							0.973932	−	0.226841i	\(-0.0728398\pi\)
\(31\)	9.82440	−	2.94123i	1.76451	−	0.528261i	0.770494	−	0.637448i	\(-0.220010\pi\)
							0.994021	+	0.109187i	\(0.0348246\pi\)
\(37\)	−2.91896	−	0.514692i	−0.479875	−	0.0846148i	−0.0715218	−	0.997439i	\(-0.522786\pi\)
							−0.408353	+	0.912824i	\(0.633897\pi\)
\(41\)	6.33245	+	2.73155i	0.988962	+	0.426597i	0.828222	−	0.560400i	\(-0.189353\pi\)
							0.160740	+	0.986997i	\(0.448612\pi\)
\(43\)	2.76962	+	1.39095i	0.422362	+	0.212118i	0.647269	−	0.762262i	\(-0.275911\pi\)
							−0.224906	+	0.974380i	\(0.572208\pi\)
\(47\)	1.72357	−	5.75712i	0.251408	−	0.839762i	−0.735367	−	0.677669i	\(-0.762990\pi\)
							0.986775	−	0.162093i	\(-0.0518245\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.131.48	yes	1872
8.3	odd	2	inner	648.2.bb.b.131.19	✓	1872
81.47	odd	54	inner	648.2.bb.b.371.19	yes	1872
648.371	even	54	inner	648.2.bb.b.371.48	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.131.19	✓	1872	8.3	odd	2	inner
648.2.bb.b.131.48	yes	1872	1.1	even	1	trivial
648.2.bb.b.371.19	yes	1872	81.47	odd	54	inner
648.2.bb.b.371.48	yes	1872	648.371	even	54	inner