Newform orbit 648.2.t.a

• Label: 648.2.t.a
• Level: $648$
• Weight: $2$
• Character orbit: 648.t
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $204$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.t (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(204\)
Relative dimension:	\(34\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 204 q + 6 q^{2} - 6 q^{4} - 12 q^{7} + 3 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−1.41009	−	0.107957i		0		1.97669	+	0.304458i	0.608623	+	1.67218i		0		0.408085	+	2.31436i	−2.75444	−	0.642710i		0		−0.677687	−	2.42362i
37.2	−1.39871	+	0.208809i		0		1.91280	−	0.584128i	1.30726	+	3.59167i		0		−0.406982	−	2.30811i	−2.55348	+	1.21644i		0		−2.57846	−	4.75075i
37.3	−1.39256	−	0.246516i		0		1.87846	+	0.686578i	−0.355965	−	0.978005i		0		−0.272085	−	1.54307i	−2.44662	−	1.41917i		0		0.254609	+	1.44968i
37.4	−1.30848	−	0.536554i		0		1.42422	+	1.40414i	−1.34230	−	3.68793i		0		−0.499959	−	2.83541i	−1.11016	−	2.60145i		0		−0.222412	+	5.54579i
37.5	−1.24516	+	0.670500i		0		1.10086	−	1.66976i	−0.882298	−	2.42409i		0		0.818935	+	4.64441i	−0.251174	+	2.81725i		0		2.72396	+	2.42681i
37.6	−1.21270	+	0.727574i		0		0.941271	−	1.76466i	0.0465753	+	0.127965i		0		−0.252128	−	1.42989i	0.142441	+	2.82484i		0		−0.149586	−	0.121295i
37.7	−1.18819	−	0.766950i		0		0.823574	+	1.82256i	0.545265	+	1.49810i		0		−0.575033	−	3.26118i	0.419254	−	2.79718i		0		0.501094	−	2.19822i
37.8	−1.12116	−	0.861977i		0		0.513991	+	1.93283i	1.15428	+	3.17134i		0		0.593321	+	3.36489i	1.08978	−	2.61005i		0		1.43950	−	4.55054i
37.9	−0.927104	+	1.06793i		0		−0.280958	−	1.98017i	−0.0465753	−	0.127965i		0		−0.252128	−	1.42989i	2.37516	+	1.53578i		0		0.179838	+	0.0688972i
37.10	−0.876534	+	1.10981i		0		−0.463377	−	1.94558i	0.882298	+	2.42409i		0		0.818935	+	4.64441i	2.56540	+	1.19110i		0		−3.46366	−	1.14561i
37.11	−0.770006	−	1.18621i		0		−0.814182	+	1.82678i	−0.857221	−	2.35519i		0		0.442985	+	2.51229i	2.79386	−	0.440839i		0		−2.13369	+	2.83036i
37.12	−0.627866	−	1.26720i		0		−1.21157	+	1.59126i	−0.405536	−	1.11420i		0		0.0413952	+	0.234764i	2.77714	+	0.536197i		0		−1.15729	+	1.21346i
37.13	−0.565802	−	1.29610i		0		−1.35974	+	1.46667i	0.890128	+	2.44561i		0		−0.478063	−	2.71123i	2.67029	+	0.932506i		0		2.66611	−	2.53742i
37.14	−0.448521	+	1.34120i		0		−1.59766	−	1.20312i	−1.30726	−	3.59167i		0		−0.406982	−	2.30811i	2.33021	−	1.60316i		0		5.40350	−	0.142366i
37.15	−0.139081	−	1.40736i		0		−1.96131	+	0.391473i	−0.141945	−	0.389992i		0		0.275847	+	1.56441i	0.823724	+	2.70582i		0		−0.529116	+	0.254008i
37.16	−0.138542	+	1.40741i		0		−1.96161	−	0.389971i	−0.608623	−	1.67218i		0		0.408085	+	2.31436i	0.820615	−	2.70677i		0		2.43776	−	0.624915i
37.17	0.000955132		1.41421i		0		−2.00000	+	0.00270152i	0.355965	+	0.978005i		0		−0.272085	−	1.54307i	−0.00573079	−	2.82842i		0		−1.38277	+	0.504344i
37.18	0.301188	+	1.38177i		0		−1.81857	+	0.832346i	1.34230	+	3.68793i		0		−0.499959	−	2.83541i	−1.69784	−	2.26215i		0		−4.69159	+	2.96551i
37.19	0.308432	−	1.38017i		0		−1.80974	−	0.851377i	0.0373388	+	0.102588i		0		0.639213	+	3.62515i	−1.73323	+	2.23516i		0		0.153105	−	0.0198926i
37.20	0.432733	−	1.34638i		0		−1.62548	−	1.16525i	−0.429535	−	1.18014i		0		−0.899466	−	5.10113i	−2.27227	+	1.68428i		0		−1.77479	+	0.0676334i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
27.e	even	9	1	inner
216.t	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.2.t.a		204
3.b	odd	2	1		216.2.t.a	✓	204
8.b	even	2	1	inner	648.2.t.a		204
12.b	even	2	1		864.2.bf.a		204
24.f	even	2	1		864.2.bf.a		204
24.h	odd	2	1		216.2.t.a	✓	204
27.e	even	9	1	inner	648.2.t.a		204
27.f	odd	18	1		216.2.t.a	✓	204
108.l	even	18	1		864.2.bf.a		204
216.t	even	18	1	inner	648.2.t.a		204
216.v	even	18	1		864.2.bf.a		204
216.x	odd	18	1		216.2.t.a	✓	204

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
216.2.t.a	✓	204	3.b	odd	2	1
216.2.t.a	✓	204	24.h	odd	2	1
216.2.t.a	✓	204	27.f	odd	18	1
216.2.t.a	✓	204	216.x	odd	18	1
648.2.t.a		204	1.a	even	1	1	trivial
648.2.t.a		204	8.b	even	2	1	inner
648.2.t.a		204	27.e	even	9	1	inner
648.2.t.a		204	216.t	even	18	1	inner
864.2.bf.a		204	12.b	even	2	1
864.2.bf.a		204	24.f	even	2	1
864.2.bf.a		204	108.l	even	18	1
864.2.bf.a		204	216.v	even	18	1

Hecke kernels

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