Embedded newform 648.2.bd.a.13.8

• Label: 648.2.bd.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	648.13
Dual form			648.2.bd.a.349.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39047 + 0.258070i) q^{2} +(-1.54866 - 0.775654i) q^{3} +(1.86680 - 0.717676i) q^{4} +(-2.80707 + 0.840380i) q^{5} +(2.35354 + 0.678857i) q^{6} +(0.791894 + 1.83582i) q^{7} +(-2.41051 + 1.47967i) q^{8} +(1.79672 + 2.40245i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39047	+	0.258070i	−0.983209	+	0.182483i
							\(3\)	−1.54866	−	0.775654i	−0.894122	−	0.447824i
\(5\)	−2.80707	+	0.840380i	−1.25536	+	0.375829i								−0.844345	−	0.535800i	\(-0.820010\pi\)
							−0.411013	+	0.911630i	\(0.634825\pi\)
\(7\)	0.791894	+	1.83582i	0.299308	+	0.693874i	0.999804	−	0.0197822i	\(-0.00629728\pi\)
							−0.700496	+	0.713656i	\(0.747038\pi\)
\(11\)	0.302258	+	1.27533i	0.0911341	+	0.384525i	0.999428	−	0.0338067i	\(-0.0107631\pi\)
							−0.908294	+	0.418332i	\(0.862615\pi\)
\(13\)	1.13176	−	1.72076i	0.313893	−	0.477252i	−0.643739	−	0.765245i	\(-0.722618\pi\)
							0.957632	+	0.287993i	\(0.0929881\pi\)
\(17\)	1.23498	−	0.449495i	0.299526	−	0.109019i	−0.187885	−	0.982191i	\(-0.560163\pi\)
							0.487412	+	0.873172i	\(0.337941\pi\)
\(19\)	0.581253	−	1.59698i	0.133349	−	0.366372i	−0.854990	−	0.518644i	\(-0.826437\pi\)
							0.988339	+	0.152272i	\(0.0486590\pi\)
\(23\)	−0.261860	+	0.607060i	−0.0546016	+	0.126581i	−0.943333	−	0.331847i	\(-0.892328\pi\)
							0.888731	+	0.458428i	\(0.151587\pi\)
\(29\)	−1.61964	+	0.0943333i	−0.300759	+	0.0175172i	−0.207861	−	0.978158i	\(-0.566650\pi\)
							−0.0928984	+	0.995676i	\(0.529613\pi\)
\(31\)	−8.37035	+	0.978354i	−1.50336	+	0.175717i	−0.827564	−	0.561372i	\(-0.810274\pi\)
							−0.675795	+	0.737089i	\(0.736200\pi\)
\(37\)	−1.90408	+	2.26919i	−0.313029	+	0.373053i	−0.899503	−	0.436915i	\(-0.856071\pi\)
							0.586474	+	0.809968i	\(0.300516\pi\)
\(41\)	3.73928	−	1.87794i	0.583978	−	0.293285i	−0.132177	−	0.991226i	\(-0.542197\pi\)
							0.716155	+	0.697941i	\(0.245900\pi\)
\(43\)	−5.51436	+	5.20253i	−0.840932	+	0.793378i	−0.980435	−	0.196843i	\(-0.936931\pi\)
							0.139503	+	0.990222i	\(0.455450\pi\)
\(47\)	−9.93236	−	1.16093i	−1.44878	−	0.169339i	−0.644965	−	0.764212i	\(-0.723128\pi\)
							−0.803819	+	0.594874i	\(0.797202\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.13.8	✓	1908
8.5	even	2	inner	648.2.bd.a.13.40	yes	1908
81.25	even	27	inner	648.2.bd.a.349.40	yes	1908
648.349	even	54	inner	648.2.bd.a.349.8	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.13.8	✓	1908	1.1	even	1	trivial
648.2.bd.a.13.40	yes	1908	8.5	even	2	inner
648.2.bd.a.349.8	yes	1908	648.349	even	54	inner
648.2.bd.a.349.40	yes	1908	81.25	even	27	inner