Embedded newform 648.2.bb.b.347.32

• Label: 648.2.bb.b.347.32
• 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			347.32
Character	\(\chi\)	\(=\)	648.347
Dual form			648.2.bb.b.155.32

$q$-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 \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.32	✓	1872	648.155	even	54	inner
648.2.bb.b.155.74	yes	1872	81.74	odd	54	inner
648.2.bb.b.347.32	yes	1872	1.1	even	1	trivial
648.2.bb.b.347.74	yes	1872	8.3	odd	2	inner