Newform orbit 648.2.bd.a

• Label: 648.2.bd.a
• Level: $648$
• Weight: $2$
• Character orbit: 648.bd
• 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:	$\mathrm{SU}(2)[C_{54}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.41420	−	0.00635899i	−1.68145	−	0.415592i	1.99992	+	0.0179858i	2.71787	−	0.813676i	2.37527	+	0.598422i	−1.20371	−	2.79050i	−2.82817	−	0.0381529i	2.65457	+	1.39760i	−3.84878	+	1.13342i
13.2	−1.41298	+	0.0591140i	1.07290	−	1.35974i	1.99301	−	0.167054i	−2.54435	+	0.761729i	−1.43560	+	1.98471i	−0.0791644	−	0.183524i	−2.80620	+	0.353858i	−0.697789	−	2.91772i	3.55008	−	1.22671i
13.3	−1.41277	−	0.0639239i	0.429774	+	1.67788i	1.99183	+	0.180619i	−4.11411	+	1.23168i	−0.499914	−	2.39793i	−1.24557	−	2.88757i	−2.80244	−	0.382498i	−2.63059	+	1.44222i	5.89101	−	1.47709i
13.4	−1.40484	−	0.162553i	−0.0416517	−	1.73155i	1.94715	+	0.456721i	1.45600	−	0.435896i	−0.222954	+	2.43932i	1.13129	+	2.62263i	−2.66120	−	0.958136i	−2.99653	+	0.144244i	−2.11630	+	0.375689i
13.5	−1.40382	+	0.171153i	−1.43648	+	0.967741i	1.94141	−	0.480537i	3.87465	−	1.15999i	1.85093	−	1.60439i	1.40145	+	3.24893i	−2.64315	+	1.00687i	1.12695	−	2.78028i	−5.24077	+	2.29158i
13.6	−1.40381	+	0.171199i	1.73135	−	0.0491944i	1.94138	−	0.480663i	0.878928	−	0.263134i	−2.42207	+	0.365466i	−1.82006	−	4.21937i	−2.64305	+	1.00712i	2.99516	−	0.170346i	−1.18880	+	0.519862i
13.7	−1.39070	+	0.256830i	1.07027	+	1.36181i	1.86808	−	0.714346i	1.22125	−	0.365619i	−1.83818	−	1.61898i	0.532782	+	1.23513i	−2.41446	+	1.47322i	−0.709034	+	2.91501i	−1.60449	+	0.822120i
13.8	−1.39047	+	0.258070i	−1.54866	−	0.775654i	1.86680	−	0.717676i	−2.80707	+	0.840380i	2.35354	+	0.678857i	0.791894	+	1.83582i	−2.41051	+	1.47967i	1.79672	+	2.40245i	3.68626	−	1.89294i
13.9	−1.35488	−	0.405333i	1.49899	+	0.867767i	1.67141	+	1.09836i	−1.99050	+	0.595916i	−1.67922	−	1.78331i	1.60061	+	3.71063i	−1.81936	−	2.16562i	1.49396	+	2.60155i	2.93843		0.000580680i
13.10	−1.35470	−	0.405938i	−1.35025	+	1.08482i	1.67043	+	1.09985i	−1.24137	+	0.371641i	2.26955	−	0.921488i	1.17620	+	2.72673i	−1.81646	−	2.16806i	0.646340	−	2.92955i	1.83254		0.000455190i
13.11	−1.34876	−	0.425256i	1.03611	+	1.38798i	1.63832	+	1.14714i	3.18189	−	0.952596i	−0.807215	−	2.31266i	−0.256454	−	0.594528i	−1.72187	−	2.24392i	−0.852968	+	2.87619i	−4.69671	−	0.0682924i
13.12	−1.33846	+	0.456643i	1.66450	−	0.479002i	1.58296	−	1.22240i	−0.183674	+	0.0549883i	−2.00913	+	1.40121i	1.23663	+	2.86682i	−1.56053	+	2.35897i	2.54112	−	1.59460i	0.220730	−	0.157473i
13.13	−1.33558	+	0.464990i	−0.266503	+	1.71143i	1.56757	−	1.24207i	0.902416	−	0.270166i	−0.439858	−	2.40967i	−0.0371225	−	0.0860596i	−1.51607	+	2.38779i	−2.85795	−	0.912201i	−1.07963	+	0.780443i
13.14	−1.31801	−	0.512678i	−1.65507	+	0.510644i	1.47432	+	1.35143i	−1.81662	+	0.543860i	2.44320	+	0.175479i	−0.762450	−	1.76756i	−1.25033	−	2.53706i	2.47849	−	1.69030i	2.67315	+	0.214525i
13.15	−1.28811	−	0.583755i	−0.393780	+	1.68669i	1.31846	+	1.50388i	1.38808	−	0.415563i	1.49185	−	1.94278i	−1.33275	−	3.08966i	−0.820426	−	2.70682i	−2.68988	−	1.32837i	−2.03059	−	0.275005i
13.16	−1.24166	+	0.676962i	−1.38930	+	1.03433i	1.08345	−	1.68111i	−0.453349	+	0.135724i	1.02484	−	2.22479i	−1.03892	−	2.40849i	−0.207223	+	2.82083i	0.860326	−	2.87399i	0.471026	−	0.475423i
13.17	−1.23745	−	0.684630i	0.393780	−	1.68669i	1.06256	+	1.69439i	−1.38808	+	0.415563i	−1.64204	+	1.81761i	−1.33275	−	3.08966i	−0.154840	−	2.82419i	−2.68988	−	1.32837i	2.00218	+	0.436081i
13.18	−1.22371	+	0.708889i	0.742903	−	1.56464i	0.994952	−	1.73496i	2.98234	−	0.892854i	0.200055	+	2.44131i	−0.865364	−	2.00614i	0.0123558	+	2.82840i	−1.89619	−	2.32475i	−3.01660	+	3.20675i
13.19	−1.21220	+	0.728408i	−0.572218	−	1.63480i	0.938845	−	1.76595i	1.28376	−	0.384332i	1.88444	+	1.56489i	0.764986	+	1.77344i	0.148264	+	2.82454i	−2.34513	+	1.87092i	−1.27622	+	1.40099i
13.20	−1.19829	−	0.751060i	1.65507	−	0.510644i	0.871818	+	1.79998i	1.81662	−	0.543860i	−2.36678	−	0.631152i	−0.762450	−	1.76756i	0.307200	−	2.81169i	2.47849	−	1.69030i	−2.58531	−	0.712685i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
81.g	even	27	1	inner
648.bd	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.bd.a	✓	1908
8.b	even	2	1	inner	648.2.bd.a	✓	1908
81.g	even	27	1	inner	648.2.bd.a	✓	1908
648.bd	even	54	1	inner	648.2.bd.a	✓	1908

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
648.2.bd.a	✓	1908	1.a	even	1	1	trivial
648.2.bd.a	✓	1908	8.b	even	2	1	inner
648.2.bd.a	✓	1908	81.g	even	27	1	inner
648.2.bd.a	✓	1908	648.bd	even	54	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(648, [\chi])\).