Embedded newform 648.2.bb.b.155.32

• Label: 648.2.bb.b.155.32
• 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			155.32
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.847711 - 1.13198i) q^{2} +(-0.462016 - 1.66929i) q^{3} +(-0.562773 + 1.91919i) q^{4} +(0.539140 - 0.354598i) q^{5} +(-1.49796 + 1.93807i) q^{6} +(0.568211 + 0.536080i) q^{7} +(2.64956 - 0.989868i) q^{8} +(-2.57308 + 1.54248i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.847711	−	1.13198i	−0.599422	−	0.800433i
							\(3\)	−0.462016	−	1.66929i	−0.266745	−	0.963767i
\(5\)	0.539140	−	0.354598i	0.241111	−	0.158581i								−0.423203	−	0.906035i	\(-0.639094\pi\)
							0.664314	+	0.747454i	\(0.268724\pi\)
\(7\)	0.568211	+	0.536080i	0.214764	+	0.202619i	0.786104	−	0.618094i	\(-0.212095\pi\)
							−0.571340	+	0.820713i	\(0.693576\pi\)
\(11\)	2.27674	+	4.53337i	0.686464	+	1.36686i	0.918994	+	0.394273i	\(0.129003\pi\)
							−0.232530	+	0.972589i	\(0.574700\pi\)
\(13\)	2.07031	−	0.893043i	0.574200	−	0.247686i	−0.0891211	−	0.996021i	\(-0.528406\pi\)
							0.663321	+	0.748335i	\(0.269147\pi\)
\(17\)	3.54620	+	4.22620i	0.860080	+	1.02500i	0.999396	+	0.0347552i	\(0.0110651\pi\)
							−0.139316	+	0.990248i	\(0.544490\pi\)
\(19\)	−0.124773	−	0.104697i	−0.0286250	−	0.0240192i	0.628363	−	0.777920i	\(-0.283725\pi\)
							−0.656988	+	0.753901i	\(0.728170\pi\)
\(23\)	1.52742	+	1.61897i	0.318488	+	0.337578i	0.866763	−	0.498721i	\(-0.166197\pi\)
							−0.548275	+	0.836298i	\(0.684715\pi\)
\(29\)	8.67805	−	1.01432i	1.61147	−	0.188354i	0.737852	−	0.674962i	\(-0.235840\pi\)
							0.873620	+	0.486608i	\(0.161766\pi\)
\(31\)	−1.62986	−	6.87692i	−0.292731	−	1.23513i	−0.898316	−	0.439350i	\(-0.855209\pi\)
							0.605585	−	0.795781i	\(-0.292939\pi\)
\(37\)	−3.35892	+	0.592269i	−0.552204	+	0.0973684i	−0.442786	−	0.896627i	\(-0.646010\pi\)
							−0.109418	+	0.993996i	\(0.534899\pi\)
\(41\)	2.81315	+	2.09432i	0.439341	+	0.327077i	0.793999	−	0.607919i	\(-0.207995\pi\)
							−0.354658	+	0.934996i	\(0.615403\pi\)
\(43\)	−0.158674	+	2.72433i	−0.0241976	+	0.415457i	0.964358	+	0.264601i	\(0.0852402\pi\)
							−0.988556	+	0.150856i	\(0.951797\pi\)
\(47\)	−10.9555	−	2.59650i	−1.59802	−	0.378738i	−0.667394	−	0.744705i	\(-0.732590\pi\)
							−0.930628	+	0.365966i	\(0.880739\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.32	✓	1872
8.3	odd	2	inner	648.2.bb.b.155.74	yes	1872
81.23	odd	54	inner	648.2.bb.b.347.74	yes	1872
648.347	even	54	inner	648.2.bb.b.347.32	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.32	✓	1872	1.1	even	1	trivial
648.2.bb.b.155.74	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.32	yes	1872	648.347	even	54	inner
648.2.bb.b.347.74	yes	1872	81.23	odd	54	inner