Embedded newform 648.2.bb.b.371.69

• Label: 648.2.bb.b.371.69
• Level: $648$
• Weight: $2$
• Character: 648.371
• 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			371.69
Character	\(\chi\)	\(=\)	648.371
Dual form			648.2.bb.b.131.69

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.647790 + 1.25713i) q^{2} +(0.952970 + 1.44632i) q^{3} +(-1.16074 + 1.62871i) q^{4} +(-0.133242 - 2.28767i) q^{5} +(-1.20089 + 2.13492i) q^{6} +(-0.141757 + 0.598121i) q^{7} +(-2.79941 - 0.404135i) q^{8} +(-1.18370 + 2.75660i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{7}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.647790	+	1.25713i	0.458056	+	0.888923i
							\(3\)	0.952970	+	1.44632i	0.550198	+	0.835034i
\(5\)	−0.133242	−	2.28767i	−0.0595874	−	1.02308i								−0.887085	−	0.461607i	\(-0.847273\pi\)
							0.827497	−	0.561470i	\(-0.189764\pi\)
\(7\)	−0.141757	+	0.598121i	−0.0535792	+	0.226068i	−0.993247	−	0.116020i	\(-0.962986\pi\)
							0.939668	+	0.342089i	\(0.111134\pi\)
\(11\)	−2.30960	+	3.51157i	−0.696369	+	1.05878i	0.298180	+	0.954510i	\(0.403620\pi\)
							−0.994549	+	0.104268i	\(0.966750\pi\)
\(13\)	−0.356940	+	3.05381i	−0.0989973	+	0.846976i	0.848934	+	0.528500i	\(0.177245\pi\)
							−0.947931	+	0.318476i	\(0.896829\pi\)
\(17\)	3.30398	+	3.93753i	0.801333	+	0.954991i	0.999684	−	0.0251396i	\(-0.00800304\pi\)
							−0.198351	+	0.980131i	\(0.563559\pi\)
\(19\)	−3.42917	−	2.87742i	−0.786706	−	0.660125i	0.158221	−	0.987404i	\(-0.449424\pi\)
							−0.944928	+	0.327279i	\(0.893868\pi\)
\(23\)	−7.04830	+	1.67048i	−1.46967	+	0.348319i	−0.886005	−	0.463675i	\(-0.846531\pi\)
							−0.583667	+	0.811993i	\(0.698382\pi\)
\(29\)	4.65878	+	6.25782i	0.865113	+	1.16205i	0.985560	+	0.169329i	\(0.0541600\pi\)
							−0.120447	+	0.992720i	\(0.538433\pi\)
\(31\)	3.35485	+	1.00438i	0.602549	+	0.180391i	0.573526	−	0.819187i	\(-0.305575\pi\)
							0.0290230	+	0.999579i	\(0.490760\pi\)
\(37\)	7.56916	−	1.33465i	1.24436	−	0.219415i	0.487578	−	0.873079i	\(-0.337880\pi\)
							0.756784	+	0.653665i	\(0.226769\pi\)
\(41\)	4.42188	−	1.90741i	0.690581	−	0.297888i	−0.0217173	−	0.999764i	\(-0.506913\pi\)
							0.712299	+	0.701876i	\(0.247654\pi\)
\(43\)	6.11067	−	3.06889i	0.931868	−	0.468002i	0.0829662	−	0.996552i	\(-0.473561\pi\)
							0.848902	+	0.528551i	\(0.177264\pi\)
\(47\)	−1.04863	−	3.50266i	−0.152958	−	0.510915i	0.846822	−	0.531876i	\(-0.178513\pi\)
							−0.999780	+	0.0209608i	\(0.993327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.371.69	yes	1872
8.3	odd	2	inner	648.2.bb.b.371.96	yes	1872
81.50	odd	54	inner	648.2.bb.b.131.96	yes	1872
648.131	even	54	inner	648.2.bb.b.131.69	✓	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.131.69	✓	1872	648.131	even	54	inner
648.2.bb.b.131.96	yes	1872	81.50	odd	54	inner
648.2.bb.b.371.69	yes	1872	1.1	even	1	trivial
648.2.bb.b.371.96	yes	1872	8.3	odd	2	inner