Newform orbit 380.3.o.a

• Label: 380.3.o.a
• Level: $380$
• Weight: $3$
• Character orbit: 380.o
• Analytic conductor: $10.354$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.o (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3542500457\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - q^{5} - 60 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
69.1		0		−2.63217	−	4.55905i		0		2.65848	−	4.23468i		0				6.09160i		0		−9.35663	+	16.2062i		0
69.2		0		−2.56966	−	4.45078i		0		−2.42657	+	4.37170i		0				1.29476i		0		−8.70628	+	15.0797i		0
69.3		0		−2.11347	−	3.66064i		0		3.33105	+	3.72882i		0			−	0.449624i		0		−4.43355	+	7.67913i		0
69.4		0		−1.86367	−	3.22797i		0		4.82957	+	1.29433i		0			−	13.6880i		0		−2.44654	+	4.23753i		0
69.5		0		−1.83229	−	3.17363i		0		−3.44448	−	3.62430i		0			−	10.5860i		0		−2.21461	+	3.83581i		0
69.6		0		−1.61189	−	2.79188i		0		−4.83370	−	1.27881i		0				4.81692i		0		−0.696399	+	1.20620i		0
69.7		0		−1.46745	−	2.54170i		0		1.82250	−	4.65602i		0				7.37072i		0		0.193162	−	0.334567i		0
69.8		0		−0.539145	−	0.933827i		0		4.21296	+	2.69277i		0				7.58035i		0		3.91864	−	6.78729i		0
69.9		0		−0.332200	−	0.575388i		0		−3.37893	+	3.68549i		0			−	0.512217i		0		4.27929	−	7.41194i		0
69.10		0		−0.136188	−	0.235884i		0		1.20560	−	4.85248i		0			−	8.79392i		0		4.46291	−	7.72998i		0
69.11		0		0.136188	+	0.235884i		0		−4.80517	−	1.38216i		0				8.79392i		0		4.46291	−	7.72998i		0
69.12		0		0.332200	+	0.575388i		0		4.88119	−	1.08349i		0				0.512217i		0		4.27929	−	7.41194i		0
69.13		0		0.539145	+	0.933827i		0		0.225529	+	4.99491i		0			−	7.58035i		0		3.91864	−	6.78729i		0
69.14		0		1.46745	+	2.54170i		0		−4.94348	−	0.749675i		0			−	7.37072i		0		0.193162	−	0.334567i		0
69.15		0		1.61189	+	2.79188i		0		1.30937	−	4.82551i		0			−	4.81692i		0		−0.696399	+	1.20620i		0
69.16		0		1.83229	+	3.17363i		0		−1.41650	−	4.79516i		0				10.5860i		0		−2.21461	+	3.83581i		0
69.17		0		1.86367	+	3.22797i		0		−1.29386	+	4.82969i		0				13.6880i		0		−2.44654	+	4.23753i		0
69.18		0		2.11347	+	3.66064i		0		1.56373	+	4.74918i		0				0.449624i		0		−4.43355	+	7.67913i		0
69.19		0		2.56966	+	4.45078i		0		4.99929	+	0.0843766i		0			−	1.29476i		0		−8.70628	+	15.0797i		0
69.20		0		2.63217	+	4.55905i		0		−4.99658	+	0.184971i		0			−	6.09160i		0		−9.35663	+	16.2062i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.d	odd	6	1	inner
95.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.3.o.a	✓	40
5.b	even	2	1	inner	380.3.o.a	✓	40
19.d	odd	6	1	inner	380.3.o.a	✓	40
95.h	odd	6	1	inner	380.3.o.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.3.o.a	✓	40	1.a	even	1	1	trivial
380.3.o.a	✓	40	5.b	even	2	1	inner
380.3.o.a	✓	40	19.d	odd	6	1	inner
380.3.o.a	✓	40	95.h	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).