Embedded newform 648.2.bb.b.155.28

• Label: 648.2.bb.b.155.28
• 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			155.28
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.900596 + 1.09038i) q^{2} +(-1.22792 - 1.22156i) q^{3} +(-0.377855 - 1.96398i) q^{4} +(2.79167 - 1.83611i) q^{5} +(2.43783 - 0.238759i) q^{6} +(1.37351 + 1.29584i) q^{7} +(2.48178 + 1.35675i) q^{8} +(0.0155586 + 2.99996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.900596	+	1.09038i	−0.636817	+	0.771015i
							\(3\)	−1.22792	−	1.22156i	−0.708938	−	0.705271i
\(5\)	2.79167	−	1.83611i	1.24847	−	0.821134i								0.259004	−	0.965876i	\(-0.416606\pi\)
							0.989469	+	0.144742i	\(0.0462354\pi\)
\(7\)	1.37351	+	1.29584i	0.519137	+	0.489780i	0.900627	−	0.434593i	\(-0.143108\pi\)
							−0.381491	+	0.924373i	\(0.624589\pi\)
\(11\)	1.73366	+	3.45199i	0.522717	+	1.04082i	0.987826	+	0.155562i	\(0.0497189\pi\)
							−0.465109	+	0.885253i	\(0.653985\pi\)
\(13\)	0.329892	−	0.142302i	0.0914957	−	0.0394674i	−0.349840	−	0.936809i	\(-0.613764\pi\)
							0.441336	+	0.897342i	\(0.354505\pi\)
\(17\)	−2.24947	−	2.68082i	−0.545578	−	0.650194i	0.420851	−	0.907130i	\(-0.361732\pi\)
							−0.966428	+	0.256936i	\(0.917287\pi\)
\(19\)	4.69816	+	3.94222i	1.07783	+	0.904408i	0.995739	−	0.0922169i	\(-0.0293953\pi\)
							0.0820924	+	0.996625i	\(0.473840\pi\)
\(23\)	2.70963	+	2.87204i	0.564998	+	0.598863i	0.945381	−	0.325966i	\(-0.105690\pi\)
							−0.380384	+	0.924829i	\(0.624208\pi\)
\(29\)	−3.12225	+	0.364939i	−0.579787	+	0.0677674i	−0.400932	−	0.916108i	\(-0.631314\pi\)
							−0.178855	+	0.983875i	\(0.557239\pi\)
\(31\)	1.70184	+	7.18063i	0.305660	+	1.28968i	0.881526	+	0.472136i	\(0.156517\pi\)
							−0.575866	+	0.817544i	\(0.695335\pi\)
\(37\)	10.0274	−	1.76809i	1.64849	−	0.290673i	0.729211	−	0.684289i	\(-0.239887\pi\)
							0.919275	+	0.393616i	\(0.128776\pi\)
\(41\)	−4.99836	−	3.72114i	−0.780612	−	0.581144i	0.131389	−	0.991331i	\(-0.458056\pi\)
							−0.912002	+	0.410186i	\(0.865464\pi\)
\(43\)	0.0897814	−	1.54149i	0.0136915	−	0.235075i	−0.984519	−	0.175278i	\(-0.943918\pi\)
							0.998211	−	0.0597969i	\(-0.0190453\pi\)
\(47\)	−9.42770	−	2.23441i	−1.37517	−	0.325922i	−0.524407	−	0.851468i	\(-0.675713\pi\)
							−0.850765	+	0.525546i	\(0.823861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.28	yes	1872
8.3	odd	2	inner	648.2.bb.b.155.15	✓	1872
81.23	odd	54	inner	648.2.bb.b.347.15	yes	1872
648.347	even	54	inner	648.2.bb.b.347.28	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.15	✓	1872	8.3	odd	2	inner
648.2.bb.b.155.28	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.15	yes	1872	81.23	odd	54	inner
648.2.bb.b.347.28	yes	1872	648.347	even	54	inner