Embedded newform 648.2.bd.a.13.1

• Label: 648.2.bd.a.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	648.13
Dual form			648.2.bd.a.349.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41420 - 0.00635899i) q^{2} +(-1.68145 - 0.415592i) q^{3} +(1.99992 + 0.0179858i) q^{4} +(2.71787 - 0.813676i) q^{5} +(2.37527 + 0.598422i) q^{6} +(-1.20371 - 2.79050i) q^{7} +(-2.82817 - 0.0381529i) q^{8} +(2.65457 + 1.39760i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41420	−	0.00635899i	−0.999990	−	0.00449649i
							\(3\)	−1.68145	−	0.415592i	−0.970787	−	0.239942i
\(5\)	2.71787	−	0.813676i	1.21547	−	0.363887i								0.386029	−	0.922486i	\(-0.373846\pi\)
							0.829437	+	0.558600i	\(0.188661\pi\)
\(7\)	−1.20371	−	2.79050i	−0.454958	−	1.05471i	−0.979155	−	0.203115i	\(-0.934894\pi\)
							0.524197	−	0.851597i	\(-0.324366\pi\)
\(11\)	−0.993864	−	4.19344i	−0.299661	−	1.26437i	−0.889532	−	0.456873i	\(-0.848969\pi\)
							0.589871	−	0.807498i	\(-0.299179\pi\)
\(13\)	0.998070	−	1.51749i	0.276815	−	0.420877i	−0.670144	−	0.742231i	\(-0.733768\pi\)
							0.946959	+	0.321354i	\(0.104138\pi\)
\(17\)	−2.06618	+	0.752026i	−0.501121	+	0.182393i	−0.580198	−	0.814475i	\(-0.697025\pi\)
							0.0790771	+	0.996869i	\(0.474803\pi\)
\(19\)	0.871549	−	2.39456i	0.199947	−	0.549350i	−0.798679	−	0.601758i	\(-0.794467\pi\)
							0.998626	+	0.0524075i	\(0.0166895\pi\)
\(23\)	−2.68457	+	6.22354i	−0.559772	+	1.29770i	0.369818	+	0.929104i	\(0.379420\pi\)
							−0.929591	+	0.368594i	\(0.879839\pi\)
\(29\)	1.81274	−	0.105580i	0.336617	−	0.0196057i	0.110996	−	0.993821i	\(-0.464596\pi\)
							0.225621	+	0.974215i	\(0.427559\pi\)
\(31\)	7.14812	−	0.835496i	1.28384	−	0.150059i	0.553331	−	0.832962i	\(-0.313357\pi\)
							0.730510	+	0.682902i	\(0.239282\pi\)
\(37\)	2.53252	−	3.01814i	0.416343	−	0.496179i	−0.516587	−	0.856234i	\(-0.672798\pi\)
							0.932931	+	0.360056i	\(0.117242\pi\)
\(41\)	−7.34424	+	3.68842i	−1.14698	+	0.576034i	−0.917679	−	0.397322i	\(-0.869939\pi\)
							−0.229298	+	0.973356i	\(0.573643\pi\)
\(43\)	−2.92676	+	2.76126i	−0.446327	+	0.421088i	−0.876309	−	0.481750i	\(-0.840001\pi\)
							0.429982	+	0.902838i	\(0.358520\pi\)
\(47\)	−4.53184	−	0.529696i	−0.661037	−	0.0772641i	−0.221042	−	0.975264i	\(-0.570946\pi\)
							−0.439995	+	0.898000i	\(0.645020\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.13.1	✓	1908
8.5	even	2	inner	648.2.bd.a.13.33	yes	1908
81.25	even	27	inner	648.2.bd.a.349.33	yes	1908
648.349	even	54	inner	648.2.bd.a.349.1	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.13.1	✓	1908	1.1	even	1	trivial
648.2.bd.a.13.33	yes	1908	8.5	even	2	inner
648.2.bd.a.349.1	yes	1908	648.349	even	54	inner
648.2.bd.a.349.33	yes	1908	81.25	even	27	inner