Embedded newform 648.2.bb.b.371.91

• Label: 648.2.bb.b.371.91
• 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			371.91
Character	\(\chi\)	\(=\)	648.371
Dual form			648.2.bb.b.131.91

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28114 + 0.598910i) q^{2} +(-0.161308 + 1.72452i) q^{3} +(1.28261 + 1.53457i) q^{4} +(0.0476203 + 0.817609i) q^{5} +(-1.23949 + 2.11274i) q^{6} +(0.895235 - 3.77729i) q^{7} +(0.724135 + 2.73416i) q^{8} +(-2.94796 - 0.556359i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28114	+	0.598910i	0.905899	+	0.423493i
							\(3\)	−0.161308	+	1.72452i	−0.0931312	+	0.995654i
\(5\)	0.0476203	+	0.817609i	0.0212964	+	0.365646i								0.992073	+	0.125661i	\(0.0401053\pi\)
							−0.970777	+	0.239984i	\(0.922858\pi\)
\(7\)	0.895235	−	3.77729i	0.338367	−	1.42768i	−0.492063	−	0.870560i	\(-0.663757\pi\)
							0.830430	−	0.557123i	\(-0.188095\pi\)
\(11\)	−2.10128	+	3.19483i	−0.633558	+	0.963279i	0.365909	+	0.930651i	\(0.380758\pi\)
							−0.999468	+	0.0326283i	\(0.989612\pi\)
\(13\)	−0.800672	+	6.85019i	−0.222066	+	1.89990i	0.178997	+	0.983850i	\(0.442715\pi\)
							−0.401063	+	0.916050i	\(0.631359\pi\)
\(17\)	−1.41251	−	1.68336i	−0.342584	−	0.408276i	0.567052	−	0.823682i	\(-0.308084\pi\)
							−0.909636	+	0.415406i	\(0.863639\pi\)
\(19\)	3.21745	+	2.69976i	0.738133	+	0.619367i	0.932335	−	0.361595i	\(-0.117768\pi\)
							−0.194203	+	0.980961i	\(0.562212\pi\)
\(23\)	1.39715	−	0.331130i	0.291326	−	0.0690454i	−0.0823545	−	0.996603i	\(-0.526244\pi\)
							0.373680	+	0.927558i	\(0.378096\pi\)
\(29\)	−5.67199	−	7.61881i	−1.05326	−	1.41478i	−0.904789	−	0.425860i	\(-0.859972\pi\)
							−0.148473	−	0.988916i	\(-0.547436\pi\)
\(31\)	5.07819	+	1.52031i	0.912070	+	0.273056i	0.708258	−	0.705954i	\(-0.249481\pi\)
							0.203813	+	0.979010i	\(0.434667\pi\)
\(37\)	1.02854	−	0.181359i	0.169090	−	0.0298152i	−0.0884619	−	0.996080i	\(-0.528195\pi\)
							0.257552	+	0.966264i	\(0.417084\pi\)
\(41\)	3.49346	−	1.50693i	0.545587	−	0.235343i	−0.105403	−	0.994430i	\(-0.533613\pi\)
							0.650990	+	0.759086i	\(0.274354\pi\)
\(43\)	4.06557	−	2.04180i	0.619993	−	0.311372i	−0.110934	−	0.993828i	\(-0.535384\pi\)
							0.730927	+	0.682455i	\(0.239088\pi\)
\(47\)	0.178805	+	0.597251i	0.0260814	+	0.0871179i	0.970039	−	0.242948i	\(-0.0781143\pi\)
							−0.943958	+	0.330066i	\(0.892929\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.371.91	yes	1872
8.3	odd	2	inner	648.2.bb.b.371.94	yes	1872
81.50	odd	54	inner	648.2.bb.b.131.94	yes	1872
648.131	even	54	inner	648.2.bb.b.131.91	✓	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.131.91	✓	1872	648.131	even	54	inner
648.2.bb.b.131.94	yes	1872	81.50	odd	54	inner
648.2.bb.b.371.91	yes	1872	1.1	even	1	trivial
648.2.bb.b.371.94	yes	1872	8.3	odd	2	inner