Newform orbit 644.2.bc.a

• Label: 644.2.bc.a
• Level: $644$
• Weight: $2$
• Character orbit: 644.bc
• Analytic conductor: $5.142$
• Analytic rank: $0$
• Dimension: $320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.bc (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.14236589017\)
Analytic rank:	\(0\)
Dimension:	\(320\)
Relative dimension:	\(16\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 320 q - 20 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1		0		−0.583248	+	3.02618i		0		0.619931	+	0.487519i		0		−2.04497	−	1.67872i		0		−6.03248	−	2.41504i		0
5.2		0		−0.558795	+	2.89930i		0		−1.41352	−	1.11160i		0		2.23525	+	1.41551i		0		−5.30859	−	2.12524i		0
5.3		0		−0.409381	+	2.12407i		0		−2.83055	−	2.22597i		0		−1.98760	−	1.74627i		0		−1.55898	−	0.624120i		0
5.4		0		−0.401668	+	2.08405i		0		0.909054	+	0.714888i		0		2.64571	+	0.0144750i		0		−1.39684	−	0.559210i		0
5.5		0		−0.277301	+	1.43878i		0		2.43721	+	1.91664i		0		−1.23059	−	2.34215i		0		0.791924	+	0.317039i		0
5.6		0		−0.248197	+	1.28777i		0		−1.66823	−	1.31191i		0		−1.56473	+	2.13345i		0		1.18836	+	0.475746i		0
5.7		0		−0.206510	+	1.07147i		0		2.27508	+	1.78914i		0		−1.75579	+	1.97920i		0		1.67969	+	0.672448i		0
5.8		0		−0.0944134	+	0.489863i		0		−0.697492	−	0.548514i		0		1.89346	−	1.84792i		0		2.55405	+	1.02249i		0
5.9		0		0.0419878	−	0.217853i		0		2.18268	+	1.71648i		0		2.01215	+	1.71792i		0		2.73941	+	1.09669i		0
5.10		0		0.134558	−	0.698152i		0		−1.09153	−	0.858389i		0		−1.45127	+	2.21220i		0		2.31579	+	0.927104i		0
5.11		0		0.280147	−	1.45354i		0		−1.86247	−	1.46466i		0		−1.88051	−	1.86110i		0		0.750811	+	0.300579i		0
5.12		0		0.281126	−	1.45862i		0		−1.67022	−	1.31348i		0		1.72584	+	2.00536i		0		0.736559	+	0.294874i		0
5.13		0		0.402820	−	2.09003i		0		1.75200	+	1.37779i		0		1.01319	−	2.44406i		0		−1.42084	−	0.568820i		0
5.14		0		0.507936	−	2.63542i		0		2.11879	+	1.66623i		0		0.210927	+	2.63733i		0		−3.90236	−	1.56227i		0
5.15		0		0.519798	−	2.69697i		0		−3.05496	−	2.40245i		0		2.63650	−	0.221078i		0		−4.21833	−	1.68877i		0
5.16		0		0.611142	−	3.17090i		0		−0.0636489	−	0.0500540i		0		−2.64575	−	0.00346821i		0		−6.89603	−	2.76075i		0
17.1		0		−2.08208	−	2.18362i		0		−3.79402	−	1.95596i		0		−1.98214	+	1.75246i		0		−0.290404	+	6.09632i		0
17.2		0		−1.98732	−	2.08424i		0		0.595169	+	0.306831i		0		−0.315825	−	2.62683i		0		−0.251873	+	5.28747i		0
17.3		0		−1.59849	−	1.67645i		0		2.22824	+	1.14874i		0		−2.60800	−	0.445329i		0		−0.112562	+	2.36298i		0
17.4		0		−1.52748	−	1.60197i		0		0.141790	+	0.0730976i		0		1.67447	+	2.04845i		0		−0.0903819	+	1.89735i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
23.d	odd	22	1	inner
161.o	even	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.2.bc.a	✓	320
7.d	odd	6	1	inner	644.2.bc.a	✓	320
23.d	odd	22	1	inner	644.2.bc.a	✓	320
161.o	even	66	1	inner	644.2.bc.a	✓	320

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.2.bc.a	✓	320	1.a	even	1	1	trivial
644.2.bc.a	✓	320	7.d	odd	6	1	inner
644.2.bc.a	✓	320	23.d	odd	22	1	inner
644.2.bc.a	✓	320	161.o	even	66	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).