Embedded newform 648.2.bb.b.155.101

• Label: 648.2.bb.b.155.101
• 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			155.101
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.101

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40044 + 0.196930i) q^{2} +(-0.739739 + 1.56614i) q^{3} +(1.92244 + 0.551575i) q^{4} +(-0.405895 + 0.266961i) q^{5} +(-1.34438 + 2.04760i) q^{6} +(0.678333 + 0.639974i) q^{7} +(2.58363 + 1.15103i) q^{8} +(-1.90557 - 2.31707i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40044	+	0.196930i	0.990257	+	0.139250i
							\(3\)	−0.739739	+	1.56614i	−0.427089	+	0.904210i
\(5\)	−0.405895	+	0.266961i	−0.181522	+	0.119389i								−0.637018	−	0.770849i	\(-0.719832\pi\)
							0.455496	+	0.890238i	\(0.349462\pi\)
\(7\)	0.678333	+	0.639974i	0.256386	+	0.241887i	0.803693	−	0.595044i	\(-0.202866\pi\)
							−0.547307	+	0.836932i	\(0.684347\pi\)
\(11\)	1.65746	+	3.30027i	0.499743	+	0.995070i	0.992191	+	0.124728i	\(0.0398059\pi\)
							−0.492448	+	0.870342i	\(0.663898\pi\)
\(13\)	−1.97624	+	0.852466i	−0.548110	+	0.236431i	−0.652082	−	0.758149i	\(-0.726104\pi\)
							0.103972	+	0.994580i	\(0.466845\pi\)
\(17\)	−0.154421	−	0.184032i	−0.0374527	−	0.0446344i	0.746993	−	0.664832i	\(-0.231497\pi\)
							−0.784446	+	0.620198i	\(0.787052\pi\)
\(19\)	2.97832	+	2.49911i	0.683275	+	0.573335i	0.916961	−	0.398977i	\(-0.130635\pi\)
							−0.233687	+	0.972312i	\(0.575079\pi\)
\(23\)	−0.582884	−	0.617821i	−0.121540	−	0.128825i	0.663746	−	0.747958i	\(-0.268966\pi\)
							−0.785286	+	0.619133i	\(0.787484\pi\)
\(29\)	−7.65039	+	0.894202i	−1.42064	+	0.166049i	−0.791465	−	0.611214i	\(-0.790681\pi\)
							−0.629176	+	0.777263i	\(0.716607\pi\)
\(31\)	−2.31766	−	9.77897i	−0.416264	−	1.75636i	−0.627203	−	0.778856i	\(-0.715800\pi\)
							0.210939	−	0.977499i	\(-0.432348\pi\)
\(37\)	6.33859	−	1.11766i	1.04206	−	0.183743i	0.373674	−	0.927560i	\(-0.378098\pi\)
							0.668383	+	0.743817i	\(0.266987\pi\)
\(41\)	6.78971	+	5.05475i	1.06037	+	0.789420i	0.978297	−	0.207206i	\(-0.0664371\pi\)
							0.0820776	+	0.996626i	\(0.473844\pi\)
\(43\)	0.111122	−	1.90789i	0.0169460	−	0.290951i	−0.979179	−	0.203000i	\(-0.934931\pi\)
							0.996125	−	0.0879515i	\(-0.0280320\pi\)
\(47\)	−6.25000	−	1.48128i	−0.911656	−	0.216066i	−0.252082	−	0.967706i	\(-0.581115\pi\)
							−0.659574	+	0.751639i	\(0.729263\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.101	yes	1872
8.3	odd	2	inner	648.2.bb.b.155.57	✓	1872
81.23	odd	54	inner	648.2.bb.b.347.57	yes	1872
648.347	even	54	inner	648.2.bb.b.347.101	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.57	✓	1872	8.3	odd	2	inner
648.2.bb.b.155.101	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.57	yes	1872	81.23	odd	54	inner
648.2.bb.b.347.101	yes	1872	648.347	even	54	inner