Embedded newform 648.2.bb.b.155.87

• Label: 648.2.bb.b.155.87
• 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			155.87
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.87

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.24468 + 0.671398i) q^{2} +(-1.72723 + 0.129091i) q^{3} +(1.09845 + 1.67135i) q^{4} +(-1.37235 + 0.902608i) q^{5} +(-2.23652 - 0.998985i) q^{6} +(1.37945 + 1.30144i) q^{7} +(0.245073 + 2.81779i) q^{8} +(2.96667 - 0.445942i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.24468	+	0.671398i	0.880121	+	0.474750i
							\(3\)	−1.72723	+	0.129091i	−0.997219	+	0.0745309i
\(5\)	−1.37235	+	0.902608i	−0.613733	+	0.403658i								−0.817968	−	0.575263i	\(-0.804900\pi\)
							0.204235	+	0.978922i	\(0.434529\pi\)
\(7\)	1.37945	+	1.30144i	0.521381	+	0.491898i	0.901344	−	0.433104i	\(-0.142582\pi\)
							−0.379963	+	0.925002i	\(0.624063\pi\)
\(11\)	0.176324	+	0.351090i	0.0531636	+	0.105858i	0.918646	−	0.395083i	\(-0.129284\pi\)
							−0.865482	+	0.500940i	\(0.832988\pi\)
\(13\)	−4.46566	+	1.92630i	−1.23855	+	0.534259i	−0.911715	−	0.410823i	\(-0.865242\pi\)
							−0.326836	+	0.945081i	\(0.605982\pi\)
\(17\)	−2.12016	−	2.52671i	−0.514215	−	0.612817i	0.444988	−	0.895537i	\(-0.353208\pi\)
							−0.959203	+	0.282719i	\(0.908763\pi\)
\(19\)	−1.51407	−	1.27046i	−0.347352	−	0.291463i	0.452374	−	0.891829i	\(-0.350577\pi\)
							−0.799726	+	0.600365i	\(0.795022\pi\)
\(23\)	0.553115	+	0.586268i	0.115333	+	0.122245i	0.782467	−	0.622692i	\(-0.213961\pi\)
							−0.667135	+	0.744937i	\(0.732479\pi\)
\(29\)	1.75252	−	0.204840i	0.325435	−	0.0380379i	0.0481931	−	0.998838i	\(-0.484654\pi\)
							0.277242	+	0.960800i	\(0.410580\pi\)
\(31\)	2.11599	+	8.92807i	0.380043	+	1.60353i	0.741374	+	0.671092i	\(0.234174\pi\)
							−0.361331	+	0.932438i	\(0.617677\pi\)
\(37\)	−3.20621	+	0.565341i	−0.527097	+	0.0929415i	−0.430863	−	0.902417i	\(-0.641791\pi\)
							−0.0962341	+	0.995359i	\(0.530680\pi\)
\(41\)	−8.88519	−	6.61478i	−1.38763	−	1.03305i	−0.993480	−	0.114010i	\(-0.963630\pi\)
							−0.394153	−	0.919045i	\(-0.628962\pi\)
\(43\)	−0.499838	+	8.58188i	−0.0762245	+	1.30872i	0.715830	+	0.698275i	\(0.246049\pi\)
							−0.792055	+	0.610450i	\(0.790989\pi\)
\(47\)	3.79499	+	0.899430i	0.553557	+	0.131195i	0.497868	−	0.867253i	\(-0.334116\pi\)
							0.0556888	+	0.998448i	\(0.482265\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.87	yes	1872
8.3	odd	2	inner	648.2.bb.b.155.46	✓	1872
81.23	odd	54	inner	648.2.bb.b.347.46	yes	1872
648.347	even	54	inner	648.2.bb.b.347.87	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.46	✓	1872	8.3	odd	2	inner
648.2.bb.b.155.87	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.46	yes	1872	81.23	odd	54	inner
648.2.bb.b.347.87	yes	1872	648.347	even	54	inner