Embedded newform 648.2.bd.a.13.20

• Label: 648.2.bd.a.13.20
• 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			13.20
Character	\(\chi\)	\(=\)	648.13
Dual form			648.2.bd.a.349.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19829 - 0.751060i) q^{2} +(1.65507 - 0.510644i) q^{3} +(0.871818 + 1.79998i) q^{4} +(1.81662 - 0.543860i) q^{5} +(-2.36678 - 0.631152i) q^{6} +(-0.762450 - 1.76756i) q^{7} +(0.307200 - 2.81169i) q^{8} +(2.47849 - 1.69030i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19829	−	0.751060i	−0.847322	−	0.531080i
							\(3\)	1.65507	−	0.510644i	0.955553	−	0.294820i
\(5\)	1.81662	−	0.543860i	0.812416	−	0.243221i								0.146464	−	0.989216i	\(-0.453211\pi\)
							0.665952	+	0.745994i	\(0.268025\pi\)
\(7\)	−0.762450	−	1.76756i	−0.288179	−	0.668074i	0.711212	−	0.702978i	\(-0.248147\pi\)
							−0.999391	+	0.0349039i	\(0.988888\pi\)
\(11\)	−1.00976	−	4.26053i	−0.304455	−	1.28460i	−0.883164	−	0.469063i	\(-0.844592\pi\)
							0.578709	−	0.815534i	\(-0.303557\pi\)
\(13\)	0.952786	−	1.44864i	0.264255	−	0.401781i	−0.678849	−	0.734278i	\(-0.737521\pi\)
							0.943104	+	0.332497i	\(0.107891\pi\)
\(17\)	−7.01411	+	2.55293i	−1.70117	+	0.619176i	−0.995958	−	0.0898180i	\(-0.971371\pi\)
							−0.705214	+	0.708994i	\(0.749149\pi\)
\(19\)	−0.813793	+	2.23588i	−0.186697	+	0.512946i	−0.997364	−	0.0725620i	\(-0.976882\pi\)
							0.810667	+	0.585507i	\(0.199105\pi\)
\(23\)	2.37547	−	5.50695i	0.495319	−	1.14828i	−0.468919	−	0.883241i	\(-0.655356\pi\)
							0.964238	−	0.265037i	\(-0.0853843\pi\)
\(29\)	4.83721	−	0.281735i	0.898247	−	0.0523169i	0.397195	−	0.917734i	\(-0.369984\pi\)
							0.501052	+	0.865417i	\(0.332947\pi\)
\(31\)	5.91909	−	0.691842i	1.06310	−	0.124258i	0.433481	−	0.901163i	\(-0.357285\pi\)
							0.629619	+	0.776904i	\(0.283211\pi\)
\(37\)	−6.87365	+	8.19170i	−1.13002	+	1.34671i	−0.199744	+	0.979848i	\(0.564011\pi\)
							−0.930277	+	0.366859i	\(0.880433\pi\)
\(41\)	4.81533	−	2.41835i	0.752028	−	0.377683i	−0.0311218	−	0.999516i	\(-0.509908\pi\)
							0.783150	+	0.621833i	\(0.213612\pi\)
\(43\)	−6.75412	+	6.37219i	−1.02999	+	0.971749i	−0.999614	−	0.0277892i	\(-0.991153\pi\)
							−0.0303802	+	0.999538i	\(0.509672\pi\)
\(47\)	8.31823	+	0.972262i	1.21334	+	0.141819i	0.698590	−	0.715522i	\(-0.253811\pi\)
							0.514749	+	0.857341i	\(0.327885\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.13.20	yes	1908
8.5	even	2	inner	648.2.bd.a.13.14	✓	1908
81.25	even	27	inner	648.2.bd.a.349.14	yes	1908
648.349	even	54	inner	648.2.bd.a.349.20	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.13.14	✓	1908	8.5	even	2	inner
648.2.bd.a.13.20	yes	1908	1.1	even	1	trivial
648.2.bd.a.349.14	yes	1908	81.25	even	27	inner
648.2.bd.a.349.20	yes	1908	648.349	even	54	inner