Newform orbit 644.3.o.a

• Label: 644.3.o.a
• Level: $644$
• Weight: $3$
• Character orbit: 644.o
• Analytic conductor: $17.548$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	644.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5477290248\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} + 22 q^{7} + 92 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
185.1		0		−4.58830	−	2.64906i		0		0.953205	−	0.550333i		0		0.0825100	−	6.99951i		0		9.53501	+	16.5151i		0
185.2		0		−4.40964	−	2.54591i		0		−6.31095	+	3.64363i		0		2.15760	−	6.65918i		0		8.46330	+	14.6589i		0
185.3		0		−4.40503	−	2.54325i		0		3.08161	−	1.77917i		0		6.79899	+	1.66547i		0		8.43619	+	14.6119i		0
185.4		0		−4.38577	−	2.53212i		0		2.56966	−	1.48359i		0		1.72808	+	6.78334i		0		8.32331	+	14.4164i		0
185.5		0		−3.96734	−	2.29055i		0		−5.27562	+	3.04588i		0		−6.03248	+	3.55094i		0		5.99321	+	10.3805i		0
185.6		0		−3.73587	−	2.15690i		0		5.75230	−	3.32109i		0		−5.17869	+	4.70969i		0		4.80447	+	8.32159i		0
185.7		0		−2.63728	−	1.52263i		0		−7.55220	+	4.36027i		0		6.98878	+	0.396249i		0		0.136833	+	0.237003i		0
185.8		0		−2.39476	−	1.38262i		0		−2.47616	+	1.42961i		0		−6.85558	+	1.41457i		0		−0.676740	−	1.17215i		0
185.9		0		−2.27303	−	1.31234i		0		7.13216	−	4.11775i		0		−3.80426	−	5.87602i		0		−1.05555	−	1.82827i		0
185.10		0		−2.26235	−	1.30617i		0		0.518018	−	0.299078i		0		−5.98441	−	3.63137i		0		−1.08786	−	1.88423i		0
185.11		0		−1.83694	−	1.06056i		0		4.26167	−	2.46048i		0		6.81145	−	1.61373i		0		−2.25043	−	3.89786i		0
185.12		0		−1.81433	−	1.04750i		0		−2.69360	+	1.55515i		0		5.53663	+	4.28320i		0		−2.30547	−	3.99319i		0
185.13		0		−1.42647	−	0.823571i		0		−2.23340	+	1.28945i		0		1.34617	−	6.86934i		0		−3.14346	−	5.44464i		0
185.14		0		−0.974707	−	0.562747i		0		6.15258	−	3.55219i		0		−2.80331	+	6.41416i		0		−3.86663	−	6.69720i		0
185.15		0		−0.473786	−	0.273540i		0		−1.87305	+	1.08140i		0		1.09669	+	6.91356i		0		−4.35035	−	7.53503i		0
185.16		0		0.711297	+	0.410668i		0		−7.78788	+	4.49634i		0		−4.20249	−	5.59813i		0		−4.16270	−	7.21001i		0
185.17		0		0.764359	+	0.441303i		0		−5.95642	+	3.43894i		0		−3.98114	+	5.75765i		0		−4.11050	−	7.11960i		0
185.18		0		0.892088	+	0.515047i		0		3.67320	−	2.12072i		0		5.78470	−	3.94173i		0		−3.96945	−	6.87529i		0
185.19		0		0.967672	+	0.558685i		0		4.84256	−	2.79585i		0		4.76468	+	5.12814i		0		−3.87574	−	6.71298i		0
185.20		0		1.41224	+	0.815357i		0		−1.22638	+	0.708053i		0		4.50393	−	5.35860i		0		−3.17039	−	5.49127i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.3.o.a	✓	60
7.d	odd	6	1	inner	644.3.o.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.3.o.a	✓	60	1.a	even	1	1	trivial
644.3.o.a	✓	60	7.d	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(644, [\chi])\).