Embedded newform 648.2.bb.b.11.15

• Label: 648.2.bb.b.11.15
• Level: $648$
• Weight: $2$
• Character: 648.11
• 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			11.15
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29842 - 0.560449i) q^{2} +(-0.296891 - 1.70642i) q^{3} +(1.37179 + 1.45540i) q^{4} +(-4.17186 - 0.487621i) q^{5} +(-0.570870 + 2.38204i) q^{6} +(-1.01368 - 2.01841i) q^{7} +(-0.965487 - 2.65854i) q^{8} +(-2.82371 + 1.01324i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29842	−	0.560449i	−0.918122	−	0.396298i
							\(3\)	−0.296891	−	1.70642i	−0.171410	−	0.985200i
\(5\)	−4.17186	−	0.487621i	−1.86571	−	0.218071i								−0.892996	−	0.450065i	\(-0.851401\pi\)
							−0.972717	+	0.231995i	\(0.925475\pi\)
\(7\)	−1.01368	−	2.01841i	−0.383136	−	0.762886i	0.616566	−	0.787303i	\(-0.288523\pi\)
							−0.999702	+	0.0244174i	\(0.992227\pi\)
\(11\)	−3.21426	−	1.38650i	−0.969136	−	0.418044i	−0.148132	−	0.988968i	\(-0.547326\pi\)
							−0.821003	+	0.570923i	\(0.806585\pi\)
\(13\)	−0.982773	−	4.14665i	−0.272572	−	1.15007i	−0.921238	−	0.388999i	\(-0.872821\pi\)
							0.648666	−	0.761073i	\(-0.275327\pi\)
\(17\)	6.97026	−	1.22904i	1.69054	−	0.298087i	0.756163	−	0.654383i	\(-0.227072\pi\)
							0.934373	+	0.356296i	\(0.115961\pi\)
\(19\)	−0.498879	+	2.82928i	−0.114451	+	0.649082i	0.872570	+	0.488489i	\(0.162452\pi\)
							−0.987021	+	0.160593i	\(0.948659\pi\)
\(23\)	−5.98800	−	3.00729i	−1.24858	−	0.627062i	−0.303171	−	0.952936i	\(-0.598045\pi\)
							−0.945413	+	0.325874i	\(0.894342\pi\)
\(29\)	1.08499	+	3.62410i	0.201477	+	0.672979i	0.997688	+	0.0679604i	\(0.0216492\pi\)
							−0.796211	+	0.605019i	\(0.793166\pi\)
\(31\)	2.49239	+	3.78950i	0.447647	+	0.680614i	0.986864	−	0.161553i	\(-0.0516502\pi\)
							−0.539217	+	0.842167i	\(0.681280\pi\)
\(37\)	−1.01387	+	2.78557i	−0.166678	+	0.457945i	−0.994708	−	0.102739i	\(-0.967239\pi\)
							0.828030	+	0.560684i	\(0.189462\pi\)
\(41\)	−5.13427	+	4.84394i	−0.801839	+	0.756496i	−0.973507	−	0.228659i	\(-0.926566\pi\)
							0.171668	+	0.985155i	\(0.445084\pi\)
\(43\)	−5.28987	−	7.10553i	−0.806698	−	1.08358i	−0.994842	−	0.101441i	\(-0.967655\pi\)
							0.188144	−	0.982141i	\(-0.439753\pi\)
\(47\)	0.317569	+	0.208868i	0.0463222	+	0.0304666i	0.572459	−	0.819933i	\(-0.305990\pi\)
							−0.526137	+	0.850400i	\(0.676360\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.15	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.39	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.39	yes	1872
648.59	even	54	inner	648.2.bb.b.59.15	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.15	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.39	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.15	yes	1872	648.59	even	54	inner
648.2.bb.b.59.39	yes	1872	81.59	odd	54	inner