Embedded newform 648.2.bb.b.347.46

• Label: 648.2.bb.b.347.46
• Level: $648$
• Weight: $2$
• Character: 648.347
• 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			347.46
Character	\(\chi\)	\(=\)	648.347
Dual form			648.2.bb.b.155.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.286215 + 1.38495i) q^{2} +(-1.72723 - 0.129091i) q^{3} +(-1.83616 - 0.792785i) q^{4} +(1.37235 + 0.902608i) q^{5} +(0.673145 - 2.35518i) q^{6} +(-1.37945 + 1.30144i) q^{7} +(1.62350 - 2.31608i) q^{8} +(2.96667 + 0.445942i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{11}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.286215	+	1.38495i	−0.202384	+	0.979306i
							\(3\)	−1.72723	−	0.129091i	−0.997219	−	0.0745309i
\(5\)	1.37235	+	0.902608i	0.613733	+	0.403658i								0.817968	−	0.575263i	\(-0.195100\pi\)
							−0.204235	+	0.978922i	\(0.565471\pi\)
\(7\)	−1.37945	+	1.30144i	−0.521381	+	0.491898i	−0.901344	−	0.433104i	\(-0.857418\pi\)
							0.379963	+	0.925002i	\(0.375937\pi\)
\(11\)	0.176324	−	0.351090i	0.0531636	−	0.105858i	−0.865482	−	0.500940i	\(-0.832988\pi\)
							0.918646	+	0.395083i	\(0.129284\pi\)
\(13\)	4.46566	+	1.92630i	1.23855	+	0.534259i	0.911715	−	0.410823i	\(-0.134758\pi\)
							0.326836	+	0.945081i	\(0.394018\pi\)
\(17\)	−2.12016	+	2.52671i	−0.514215	+	0.612817i	−0.959203	−	0.282719i	\(-0.908763\pi\)
							0.444988	+	0.895537i	\(0.353208\pi\)
\(19\)	−1.51407	+	1.27046i	−0.347352	+	0.291463i	−0.799726	−	0.600365i	\(-0.795022\pi\)
							0.452374	+	0.891829i	\(0.350577\pi\)
\(23\)	−0.553115	+	0.586268i	−0.115333	+	0.122245i	−0.782467	−	0.622692i	\(-0.786039\pi\)
							0.667135	+	0.744937i	\(0.267521\pi\)
\(29\)	−1.75252	−	0.204840i	−0.325435	−	0.0380379i	−0.0481931	−	0.998838i	\(-0.515346\pi\)
							−0.277242	+	0.960800i	\(0.589420\pi\)
\(31\)	−2.11599	+	8.92807i	−0.380043	+	1.60353i	0.361331	+	0.932438i	\(0.382323\pi\)
							−0.741374	+	0.671092i	\(0.765826\pi\)
\(37\)	3.20621	+	0.565341i	0.527097	+	0.0929415i	0.430863	−	0.902417i	\(-0.358209\pi\)
							0.0962341	+	0.995359i	\(0.469320\pi\)
\(41\)	−8.88519	+	6.61478i	−1.38763	+	1.03305i	−0.394153	+	0.919045i	\(0.628962\pi\)
							−0.993480	+	0.114010i	\(0.963630\pi\)
\(43\)	−0.499838	−	8.58188i	−0.0762245	−	1.30872i	−0.792055	−	0.610450i	\(-0.790989\pi\)
							0.715830	−	0.698275i	\(-0.246049\pi\)
\(47\)	−3.79499	+	0.899430i	−0.553557	+	0.131195i	−0.497868	−	0.867253i	\(-0.665884\pi\)
							−0.0556888	+	0.998448i	\(0.517735\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.46	✓	1872	648.155	even	54	inner
648.2.bb.b.155.87	yes	1872	81.74	odd	54	inner
648.2.bb.b.347.46	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.87	yes	1872	8.3	odd	2	inner