Embedded newform 648.2.bd.a.13.9

• Label: 648.2.bd.a.13.9
• Level: $648$
• Weight: $2$
• Character: 648.13
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bd (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1908\)
Relative dimension:	\(106\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			13.9
Character	\(\chi\)	\(=\)	648.13
Dual form			648.2.bd.a.349.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35488 - 0.405333i) q^{2} +(1.49899 + 0.867767i) q^{3} +(1.67141 + 1.09836i) q^{4} +(-1.99050 + 0.595916i) q^{5} +(-1.67922 - 1.78331i) q^{6} +(1.60061 + 3.71063i) q^{7} +(-1.81936 - 2.16562i) q^{8} +(1.49396 + 2.60155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{4}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35488	−	0.405333i	−0.958046	−	0.286614i
							\(3\)	1.49899	+	0.867767i	0.865444	+	0.501006i
\(5\)	−1.99050	+	0.595916i	−0.890178	+	0.266502i								−0.699058	−	0.715065i	\(-0.746397\pi\)
							−0.191120	+	0.981567i	\(0.561212\pi\)
\(7\)	1.60061	+	3.71063i	0.604973	+	1.40249i	0.895259	+	0.445546i	\(0.146990\pi\)
							−0.290286	+	0.956940i	\(0.593750\pi\)
\(11\)	0.0645003	+	0.272148i	0.0194476	+	0.0820557i	0.981903	−	0.189383i	\(-0.0606488\pi\)
							−0.962456	+	0.271439i	\(0.912501\pi\)
\(13\)	−0.539041	+	0.819573i	−0.149503	+	0.227309i	−0.902473	−	0.430747i	\(-0.858250\pi\)
							0.752970	+	0.658055i	\(0.228621\pi\)
\(17\)	−5.18177	+	1.88601i	−1.25676	+	0.457425i	−0.882682	−	0.469971i	\(-0.844264\pi\)
							−0.374082	+	0.927396i	\(0.622042\pi\)
\(19\)	2.90720	−	7.98747i	0.666958	−	1.83245i	0.124786	−	0.992184i	\(-0.460176\pi\)
							0.542172	−	0.840267i	\(-0.317602\pi\)
\(23\)	−1.67746	+	3.88878i	−0.349774	+	0.810866i	0.648973	+	0.760812i	\(0.275199\pi\)
							−0.998746	+	0.0500549i	\(0.984060\pi\)
\(29\)	4.52472	−	0.263535i	0.840219	−	0.0489372i	0.367364	−	0.930077i	\(-0.380260\pi\)
							0.472855	+	0.881140i	\(0.343223\pi\)
\(31\)	−7.03951	+	0.822801i	−1.26433	+	0.147779i	−0.721710	−	0.692196i	\(-0.756643\pi\)
							−0.542624	+	0.839975i	\(0.682569\pi\)
\(37\)	−4.64852	+	5.53989i	−0.764211	+	0.910752i	−0.998106	−	0.0615101i	\(-0.980408\pi\)
							0.233895	+	0.972262i	\(0.424853\pi\)
\(41\)	6.04750	−	3.03717i	0.944461	−	0.474326i	0.0912130	−	0.995831i	\(-0.470926\pi\)
							0.853248	+	0.521505i	\(0.174629\pi\)
\(43\)	1.38689	−	1.30846i	0.211498	−	0.199538i	−0.573201	−	0.819415i	\(-0.694299\pi\)
							0.784700	+	0.619876i	\(0.212817\pi\)
\(47\)	4.24193	+	0.495811i	0.618749	+	0.0723214i	0.419692	−	0.907666i	\(-0.362138\pi\)
							0.199057	+	0.979988i	\(0.436212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.13.9	✓	1908
8.5	even	2	inner	648.2.bd.a.13.24	yes	1908
81.25	even	27	inner	648.2.bd.a.349.24	yes	1908
648.349	even	54	inner	648.2.bd.a.349.9	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.13.9	✓	1908	1.1	even	1	trivial
648.2.bd.a.13.24	yes	1908	8.5	even	2	inner
648.2.bd.a.349.9	yes	1908	648.349	even	54	inner
648.2.bd.a.349.24	yes	1908	81.25	even	27	inner