Newform orbit 648.2.bb.a

• Label: 648.2.bb.a
• Level: $648$
• Weight: $2$
• Character orbit: 648.bb
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $36$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(2\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{54}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 36 q+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.02866	−	0.970492i	0.537215	−	1.64663i	0.116290	+	1.99662i		0		−2.15066	+	1.17247i		0		1.81808	−	2.16670i	−2.42280	−	1.76919i		0
11.2	1.02866	+	0.970492i	−1.73153	+	0.0423866i	0.116290	+	1.99662i		0		−1.82230	−	1.63684i		0		−1.81808	+	2.16670i	2.99641	−	0.146787i		0
59.1	−1.02866	+	0.970492i	0.537215	+	1.64663i	0.116290	−	1.99662i		0		−2.15066	−	1.17247i		0		1.81808	+	2.16670i	−2.42280	+	1.76919i		0
59.2	1.02866	−	0.970492i	−1.73153	−	0.0423866i	0.116290	−	1.99662i		0		−1.82230	+	1.63684i		0		−1.81808	−	2.16670i	2.99641	+	0.146787i		0
83.1	−1.41182	−	0.0822292i	−1.06800	+	1.36359i	1.98648	+	0.232186i		0		1.61995	−	1.83732i		0		−2.78546	−	0.491151i	−0.718758	−	2.91263i		0
83.2	1.41182	+	0.0822292i	1.64160	+	0.552388i	1.98648	+	0.232186i		0		2.27223	+	0.914862i		0		2.78546	+	0.491151i	2.38973	+	1.81361i		0
131.1	−1.29855	+	0.560141i	−0.258935	−	1.71259i	1.37248	−	1.45475i		0		1.29553	+	2.07885i		0		−0.967379	+	2.65785i	−2.86591	+	0.886896i		0
131.2	1.29855	−	0.560141i	−1.52833	+	0.814988i	1.37248	−	1.45475i		0		−1.52811	+	1.91439i		0		0.967379	−	2.65785i	1.67159	−	2.49114i		0
155.1	−1.13437	+	0.844510i	1.46997	−	0.916079i	0.573606	−	1.91598i		0		−0.893853	+	2.28058i		0		0.967379	+	2.65785i	1.32160	−	2.69321i		0
155.2	1.13437	−	0.844510i	−1.35368	−	1.08054i	0.573606	−	1.91598i		0		−2.44810	−	0.0825402i		0		−0.967379	−	2.65785i	0.664878	+	2.92540i		0
203.1	−1.41182	+	0.0822292i	−1.06800	−	1.36359i	1.98648	−	0.232186i		0		1.61995	+	1.83732i		0		−2.78546	+	0.491151i	−0.718758	+	2.91263i		0
203.2	1.41182	−	0.0822292i	1.64160	−	0.552388i	1.98648	−	0.232186i		0		2.27223	−	0.914862i		0		2.78546	−	0.491151i	2.38973	−	1.81361i		0
227.1	−0.326140	+	1.37609i	−1.69463	+	0.358075i	−1.78727	−	0.897598i		0		0.0599437	−	2.44876i		0		1.81808	−	2.16670i	2.74356	−	1.21361i		0
227.2	0.326140	−	1.37609i	0.902474	+	1.47836i	−1.78727	−	0.897598i		0		2.32869	−	0.759737i		0		−1.81808	+	2.16670i	−1.37108	+	2.66836i		0
275.1	−1.35480	−	0.405601i	0.829058	+	1.52074i	1.67098	+	1.09902i		0		−0.506394	−	2.39657i		0		−1.81808	−	2.16670i	−1.62533	+	2.52157i		0
275.2	1.35480	+	0.405601i	1.15742	−	1.28856i	1.67098	+	1.09902i		0		2.09071	−	1.27629i		0		1.81808	+	2.16670i	−0.320765	−	2.98280i		0
299.1	−0.634698	−	1.26379i	−0.342420	−	1.69787i	−1.19432	+	1.60425i		0		−1.92841	+	1.51038i		0		2.78546	+	0.491151i	−2.76550	+	1.16277i		0
299.2	0.634698	+	1.26379i	1.71490	+	0.243119i	−1.19432	+	1.60425i		0		0.781195	+	2.32158i		0		−2.78546	−	0.491151i	2.88179	+	0.833850i		0
347.1	−1.13437	−	0.844510i	1.46997	+	0.916079i	0.573606	+	1.91598i		0		−0.893853	−	2.28058i		0		0.967379	−	2.65785i	1.32160	+	2.69321i		0
347.2	1.13437	+	0.844510i	−1.35368	+	1.08054i	0.573606	+	1.91598i		0		−2.44810	+	0.0825402i		0		−0.967379	+	2.65785i	0.664878	−	2.92540i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
81.h	odd	54	1	inner
648.bb	even	54	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.2.bb.a	✓	36
8.d	odd	2	1	CM	648.2.bb.a	✓	36
81.h	odd	54	1	inner	648.2.bb.a	✓	36
648.bb	even	54	1	inner	648.2.bb.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
648.2.bb.a	✓	36	1.a	even	1	1	trivial
648.2.bb.a	✓	36	8.d	odd	2	1	CM
648.2.bb.a	✓	36	81.h	odd	54	1	inner
648.2.bb.a	✓	36	648.bb	even	54	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\).