Embedded newform 648.2.bb.b.11.6

• Label: 648.2.bb.b.11.6
• Level: $648$
• Weight: $2$
• Character: 648.11
• 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			11.6
Character	\(\chi\)	\(=\)	648.11
Dual form			648.2.bb.b.59.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.40154 - 0.188885i) q^{2} +(0.0948694 + 1.72945i) q^{3} +(1.92864 + 0.529461i) q^{4} +(3.02884 + 0.354021i) q^{5} +(0.193704 - 2.44182i) q^{6} +(-0.170146 - 0.338788i) q^{7} +(-2.60307 - 1.10635i) q^{8} +(-2.98200 + 0.328144i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.40154	−	0.188885i	−0.991040	−	0.133562i
							\(3\)	0.0948694	+	1.72945i	0.0547729	+	0.998499i
\(5\)	3.02884	+	0.354021i	1.35454	+	0.158323i								0.762187	−	0.647356i	\(-0.224125\pi\)
							0.592352	+	0.805679i	\(0.298200\pi\)
\(7\)	−0.170146	−	0.338788i	−0.0643091	−	0.128050i	0.859182	−	0.511671i	\(-0.170973\pi\)
							−0.923491	+	0.383621i	\(0.874677\pi\)
\(11\)	5.01386	+	2.16277i	1.51174	+	0.652099i	0.980748	−	0.195280i	\(-0.0625616\pi\)
							0.530988	+	0.847379i	\(0.321821\pi\)
\(13\)	0.733375	+	3.09435i	0.203402	+	0.858220i	0.975017	+	0.222130i	\(0.0713010\pi\)
							−0.771615	+	0.636090i	\(0.780551\pi\)
\(17\)	−4.56447	+	0.804840i	−1.10705	+	0.195202i	−0.697147	−	0.716928i	\(-0.745548\pi\)
							−0.409900	+	0.912130i	\(0.634436\pi\)
\(19\)	1.20736	−	6.84725i	0.276986	−	1.57087i	−0.455594	−	0.890188i	\(-0.650573\pi\)
							0.732581	−	0.680680i	\(-0.238316\pi\)
\(23\)	7.09612	+	3.56380i	1.47964	+	0.743105i	0.991555	−	0.129690i	\(-0.0413982\pi\)
							0.488088	+	0.872794i	\(0.337694\pi\)
\(29\)	1.24521	+	4.15928i	0.231229	+	0.772358i	0.992318	+	0.123711i	\(0.0394796\pi\)
							−0.761089	+	0.648647i	\(0.775335\pi\)
\(31\)	−2.80369	−	4.26280i	−0.503557	−	0.765622i	0.490849	−	0.871245i	\(-0.336687\pi\)
							−0.994406	+	0.105623i	\(0.966316\pi\)
\(37\)	−1.40879	+	3.87062i	−0.231604	+	0.636326i	−0.999993	−	0.00365631i	\(-0.998836\pi\)
							0.768390	+	0.639982i	\(0.221058\pi\)
\(41\)	−1.25922	+	1.18801i	−0.196657	+	0.185536i	−0.778271	−	0.627929i	\(-0.783903\pi\)
							0.581614	+	0.813465i	\(0.302421\pi\)
\(43\)	−3.99102	−	5.36087i	−0.608625	−	0.817526i	0.385843	−	0.922565i	\(-0.373911\pi\)
							−0.994468	+	0.105039i	\(0.966503\pi\)
\(47\)	−1.07057	−	0.704124i	−0.156158	−	0.102707i	0.469025	−	0.883185i	\(-0.344605\pi\)
							−0.625183	+	0.780478i	\(0.714976\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.11.6	✓	1872
8.3	odd	2	inner	648.2.bb.b.11.46	yes	1872
81.59	odd	54	inner	648.2.bb.b.59.46	yes	1872
648.59	even	54	inner	648.2.bb.b.59.6	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.11.6	✓	1872	1.1	even	1	trivial
648.2.bb.b.11.46	yes	1872	8.3	odd	2	inner
648.2.bb.b.59.6	yes	1872	648.59	even	54	inner
648.2.bb.b.59.46	yes	1872	81.59	odd	54	inner