Embedded newform 648.2.bb.b.155.23

• Label: 648.2.bb.b.155.23
• Level: $648$
• Weight: $2$
• Character: 648.155
• 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			155.23
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.10347 + 0.884504i) q^{2} +(-1.15307 + 1.29246i) q^{3} +(0.435304 - 1.95205i) q^{4} +(-3.47249 + 2.28389i) q^{5} +(0.129194 - 2.44608i) q^{6} +(-0.348278 - 0.328583i) q^{7} +(1.24625 + 2.53907i) q^{8} +(-0.340880 - 2.98057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{43}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.10347	+	0.884504i	−0.780273	+	0.625439i
							\(3\)	−1.15307	+	1.29246i	−0.665723	+	0.746199i
\(5\)	−3.47249	+	2.28389i	−1.55295	+	1.02139i								−0.573615	+	0.819125i	\(0.694459\pi\)
							−0.979331	+	0.202263i	\(0.935170\pi\)
\(7\)	−0.348278	−	0.328583i	−0.131637	−	0.124193i	0.617608	−	0.786486i	\(-0.288102\pi\)
							−0.749245	+	0.662293i	\(0.769583\pi\)
\(11\)	−0.145259	−	0.289234i	−0.0437972	−	0.0872073i	0.870670	−	0.491868i	\(-0.163686\pi\)
							−0.914467	+	0.404661i	\(0.867390\pi\)
\(13\)	−3.51725	+	1.51719i	−0.975509	+	0.420793i	−0.823332	−	0.567560i	\(-0.807888\pi\)
							−0.152177	+	0.988353i	\(0.548628\pi\)
\(17\)	−0.326220	−	0.388774i	−0.0791200	−	0.0942916i	0.725032	−	0.688715i	\(-0.241825\pi\)
							−0.804152	+	0.594423i	\(0.797380\pi\)
\(19\)	−5.23137	−	4.38964i	−1.20016	−	1.00705i	−0.999625	−	0.0273662i	\(-0.991288\pi\)
							−0.200534	−	0.979687i	\(-0.564268\pi\)
\(23\)	3.08248	+	3.26724i	0.642742	+	0.681267i	0.964054	−	0.265707i	\(-0.0856052\pi\)
							−0.321312	+	0.946973i	\(0.604124\pi\)
\(29\)	−1.89851	+	0.221904i	−0.352544	+	0.0412065i	−0.290522	−	0.956868i	\(-0.593829\pi\)
							−0.0620221	+	0.998075i	\(0.519755\pi\)
\(31\)	1.59267	+	6.72001i	0.286052	+	1.20695i	0.906331	+	0.422568i	\(0.138871\pi\)
							−0.620279	+	0.784381i	\(0.712981\pi\)
\(37\)	9.87455	−	1.74115i	1.62337	−	0.286243i	0.713347	−	0.700811i	\(-0.247178\pi\)
							0.910018	+	0.414568i	\(0.136067\pi\)
\(41\)	5.36046	+	3.99071i	0.837163	+	0.623244i	0.927983	−	0.372623i	\(-0.121541\pi\)
							−0.0908203	+	0.995867i	\(0.528949\pi\)
\(43\)	−0.0963426	+	1.65414i	−0.0146921	+	0.252254i	0.982963	+	0.183806i	\(0.0588417\pi\)
							−0.997655	+	0.0684479i	\(0.978195\pi\)
\(47\)	−4.08981	−	0.969302i	−0.596560	−	0.141387i	−0.0787665	−	0.996893i	\(-0.525098\pi\)
							−0.517793	+	0.855506i	\(0.673246\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.23	yes	1872
8.3	odd	2	inner	648.2.bb.b.155.21	✓	1872
81.23	odd	54	inner	648.2.bb.b.347.21	yes	1872
648.347	even	54	inner	648.2.bb.b.347.23	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.21	✓	1872	8.3	odd	2	inner
648.2.bb.b.155.23	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.21	yes	1872	81.23	odd	54	inner
648.2.bb.b.347.23	yes	1872	648.347	even	54	inner