Embedded newform 648.2.bd.a.349.11

• Label: 648.2.bd.a.349.11
• Level: $648$
• Weight: $2$
• Character: 648.349
• 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			349.11
Character	\(\chi\)	\(=\)	648.349
Dual form			648.2.bd.a.13.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34876 + 0.425256i) q^{2} +(1.03611 - 1.38798i) q^{3} +(1.63832 - 1.14714i) q^{4} +(3.18189 + 0.952596i) q^{5} +(-0.807215 + 2.31266i) q^{6} +(-0.256454 + 0.594528i) q^{7} +(-1.72187 + 2.24392i) q^{8} +(-0.852968 - 2.87619i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{23}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34876	+	0.425256i	−0.953718	+	0.300701i
							\(3\)	1.03611	−	1.38798i	0.598196	−	0.801350i
\(5\)	3.18189	+	0.952596i	1.42299	+	0.426014i								0.903545	−	0.428493i	\(-0.140955\pi\)
							0.519441	+	0.854507i	\(0.326140\pi\)
\(7\)	−0.256454	+	0.594528i	−0.0969306	+	0.224710i	−0.959781	−	0.280748i	\(-0.909417\pi\)
							0.862851	+	0.505459i	\(0.168677\pi\)
\(11\)	0.966294	−	4.07711i	0.291349	−	1.22930i	−0.608662	−	0.793429i	\(-0.708294\pi\)
							0.900011	−	0.435867i	\(-0.143558\pi\)
\(13\)	3.56734	+	5.42388i	0.989403	+	1.50431i	0.858431	+	0.512929i	\(0.171440\pi\)
							0.130972	+	0.991386i	\(0.458190\pi\)
\(17\)	1.51871	+	0.552765i	0.368341	+	0.134065i	0.519558	−	0.854435i	\(-0.326097\pi\)
							−0.151216	+	0.988501i	\(0.548319\pi\)
\(19\)	0.333005	+	0.914924i	0.0763966	+	0.209898i	0.972012	−	0.234931i	\(-0.0754865\pi\)
							−0.895615	+	0.444829i	\(0.853264\pi\)
\(23\)	−0.276621	−	0.641279i	−0.0576794	−	0.133716i	0.886939	−	0.461886i	\(-0.152827\pi\)
							−0.944619	+	0.328170i	\(0.893568\pi\)
\(29\)	−8.74007	−	0.509051i	−1.62299	−	0.0945284i	−0.777629	−	0.628723i	\(-0.783578\pi\)
							−0.845361	+	0.534195i	\(0.820615\pi\)
\(31\)	5.76386	+	0.673699i	1.03522	+	0.121000i	0.616693	−	0.787204i	\(-0.288472\pi\)
							0.418528	+	0.908204i	\(0.362546\pi\)
\(37\)	−0.420838	−	0.501536i	−0.0691854	−	0.0824519i	0.730342	−	0.683082i	\(-0.239361\pi\)
							−0.799527	+	0.600630i	\(0.794916\pi\)
\(41\)	−1.61970	−	0.813443i	−0.252954	−	0.127038i	0.317804	−	0.948156i	\(-0.397054\pi\)
							−0.570759	+	0.821118i	\(0.693351\pi\)
\(43\)	−2.18627	−	2.06264i	−0.333403	−	0.314550i	0.501275	−	0.865288i	\(-0.332865\pi\)
							−0.834678	+	0.550739i	\(0.814346\pi\)
\(47\)	−5.52367	+	0.645625i	−0.805710	+	0.0941740i	−0.508970	−	0.860784i	\(-0.669974\pi\)
							−0.296740	+	0.954958i	\(0.595900\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.349.11	yes	1908
8.5	even	2	inner	648.2.bd.a.349.22	yes	1908
81.13	even	27	inner	648.2.bd.a.13.22	yes	1908
648.13	even	54	inner	648.2.bd.a.13.11	✓	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.13.11	✓	1908	648.13	even	54	inner
648.2.bd.a.13.22	yes	1908	81.13	even	27	inner
648.2.bd.a.349.11	yes	1908	1.1	even	1	trivial
648.2.bd.a.349.22	yes	1908	8.5	even	2	inner