Embedded newform 648.2.bb.b.155.2

• Label: 648.2.bb.b.155.2
• 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			155.2
Character	\(\chi\)	\(=\)	648.155
Dual form			648.2.bb.b.347.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41415 - 0.0136918i) q^{2} +(-1.36990 + 1.05989i) q^{3} +(1.99963 + 0.0387243i) q^{4} +(0.991758 - 0.652289i) q^{5} +(1.95176 - 1.48008i) q^{6} +(0.145385 + 0.137164i) q^{7} +(-2.82723 - 0.0821402i) q^{8} +(0.753269 - 2.90389i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1872 q - 18 q^{2} - 36 q^{3} - 18 q^{4} - 18 q^{6} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41415	−	0.0136918i	−0.999953	−	0.00968153i
							\(3\)	−1.36990	+	1.05989i	−0.790914	+	0.611927i
\(5\)	0.991758	−	0.652289i	0.443527	−	0.291713i								−0.308034	−	0.951375i	\(-0.599671\pi\)
							0.751562	+	0.659663i	\(0.229301\pi\)
\(7\)	0.145385	+	0.137164i	0.0549505	+	0.0518431i	0.713227	−	0.700933i	\(-0.247233\pi\)
							−0.658277	+	0.752776i	\(0.728714\pi\)
\(11\)	−0.414045	−	0.824431i	−0.124839	−	0.248575i	0.822528	−	0.568725i	\(-0.192563\pi\)
							−0.947367	+	0.320150i	\(0.896267\pi\)
\(13\)	−5.37106	+	2.31685i	−1.48966	+	0.642578i	−0.976462	−	0.215690i	\(-0.930800\pi\)
							−0.513202	+	0.858268i	\(0.671541\pi\)
\(17\)	−3.81791	−	4.55001i	−0.925979	−	1.10354i	−0.994379	−	0.105881i	\(-0.966234\pi\)
							0.0683994	−	0.997658i	\(-0.478211\pi\)
\(19\)	3.38690	+	2.84195i	0.777008	+	0.651987i	0.942493	−	0.334225i	\(-0.108475\pi\)
							−0.165485	+	0.986212i	\(0.552919\pi\)
\(23\)	2.12189	+	2.24907i	0.442445	+	0.468964i	0.909765	−	0.415124i	\(-0.136262\pi\)
							−0.467320	+	0.884088i	\(0.654780\pi\)
\(29\)	−4.36017	+	0.509631i	−0.809664	+	0.0946361i	−0.510844	−	0.859673i	\(-0.670667\pi\)
							−0.298820	+	0.954310i	\(0.596593\pi\)
\(31\)	−0.0796145	−	0.335920i	−0.0142992	−	0.0603330i	0.965428	−	0.260672i	\(-0.0839440\pi\)
							−0.979727	+	0.200339i	\(0.935796\pi\)
\(37\)	−6.35133	+	1.11991i	−1.04415	+	0.184112i	−0.669315	−	0.742979i	\(-0.733412\pi\)
							−0.374837	+	0.927091i	\(0.622301\pi\)
\(41\)	−3.38407	−	2.51935i	−0.528504	−	0.393456i	0.299571	−	0.954074i	\(-0.403156\pi\)
							−0.828075	+	0.560618i	\(0.810564\pi\)
\(43\)	−0.194060	+	3.33187i	−0.0295938	+	0.508106i	0.950644	+	0.310284i	\(0.100424\pi\)
							−0.980238	+	0.197823i	\(0.936613\pi\)
\(47\)	−9.46275	−	2.24271i	−1.38028	−	0.327133i	−0.527570	−	0.849512i	\(-0.676897\pi\)
							−0.852714	+	0.522378i	\(0.825045\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bb.b.155.2	✓	1872
8.3	odd	2	inner	648.2.bb.b.155.43	yes	1872
81.23	odd	54	inner	648.2.bb.b.347.43	yes	1872
648.347	even	54	inner	648.2.bb.b.347.2	yes	1872

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bb.b.155.2	✓	1872	1.1	even	1	trivial
648.2.bb.b.155.43	yes	1872	8.3	odd	2	inner
648.2.bb.b.347.2	yes	1872	648.347	even	54	inner
648.2.bb.b.347.43	yes	1872	81.23	odd	54	inner