Newform orbit 384.3.m.a

• Label: 384.3.m.a
• Level: $384$
• Weight: $3$
• Character orbit: 384.m
• Analytic conductor: $10.463$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.m (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4632421514\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(16\) over \(\Q(\zeta_{8})\)
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
79.1		0		−0.662827	−	1.60021i		0		−3.03571	+	7.32885i		0		4.76330	−	4.76330i		0		−2.12132	+	2.12132i		0
79.2		0		−0.662827	−	1.60021i		0		−2.48925	+	6.00957i		0		−5.51311	+	5.51311i		0		−2.12132	+	2.12132i		0
79.3		0		−0.662827	−	1.60021i		0		−1.28772	+	3.10884i		0		−0.299204	+	0.299204i		0		−2.12132	+	2.12132i		0
79.4		0		−0.662827	−	1.60021i		0		−0.574930	+	1.38800i		0		7.38174	−	7.38174i		0		−2.12132	+	2.12132i		0
79.5		0		−0.662827	−	1.60021i		0		−0.141605	+	0.341864i		0		−1.60101	+	1.60101i		0		−2.12132	+	2.12132i		0
79.6		0		−0.662827	−	1.60021i		0		1.43378	−	3.46145i		0		−5.32763	+	5.32763i		0		−2.12132	+	2.12132i		0
79.7		0		−0.662827	−	1.60021i		0		2.73167	−	6.59484i		0		−7.96095	+	7.96095i		0		−2.12132	+	2.12132i		0
79.8		0		−0.662827	−	1.60021i		0		3.36376	−	8.12084i		0		8.55687	−	8.55687i		0		−2.12132	+	2.12132i		0
79.9		0		0.662827	+	1.60021i		0		−3.53145	+	8.52568i		0		−2.86767	+	2.86767i		0		−2.12132	+	2.12132i		0
79.10		0		0.662827	+	1.60021i		0		−1.48548	+	3.58626i		0		7.09685	−	7.09685i		0		−2.12132	+	2.12132i		0
79.11		0		0.662827	+	1.60021i		0		−0.872612	+	2.10667i		0		6.43807	−	6.43807i		0		−2.12132	+	2.12132i		0
79.12		0		0.662827	+	1.60021i		0		−0.862642	+	2.08260i		0		−6.00123	+	6.00123i		0		−2.12132	+	2.12132i		0
79.13		0		0.662827	+	1.60021i		0		−0.0405573	+	0.0979141i		0		−2.00065	+	2.00065i		0		−2.12132	+	2.12132i		0
79.14		0		0.662827	+	1.60021i		0		1.70666	−	4.12024i		0		−2.50052	+	2.50052i		0		−2.12132	+	2.12132i		0
79.15		0		0.662827	+	1.60021i		0		2.47553	−	5.97646i		0		−3.72058	+	3.72058i		0		−2.12132	+	2.12132i		0
79.16		0		0.662827	+	1.60021i		0		2.61055	−	6.30243i		0		3.55574	−	3.55574i		0		−2.12132	+	2.12132i		0
175.1		0		−0.662827	+	1.60021i		0		−3.03571	−	7.32885i		0		4.76330	+	4.76330i		0		−2.12132	−	2.12132i		0
175.2		0		−0.662827	+	1.60021i		0		−2.48925	−	6.00957i		0		−5.51311	−	5.51311i		0		−2.12132	−	2.12132i		0
175.3		0		−0.662827	+	1.60021i		0		−1.28772	−	3.10884i		0		−0.299204	−	0.299204i		0		−2.12132	−	2.12132i		0
175.4		0		−0.662827	+	1.60021i		0		−0.574930	−	1.38800i		0		7.38174	+	7.38174i		0		−2.12132	−	2.12132i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
32.h	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	384.3.m.a		64
4.b	odd	2	1		96.3.m.a	✓	64
12.b	even	2	1		288.3.u.b		64
32.g	even	8	1		96.3.m.a	✓	64
32.h	odd	8	1	inner	384.3.m.a		64
96.p	odd	8	1		288.3.u.b		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.3.m.a	✓	64	4.b	odd	2	1
96.3.m.a	✓	64	32.g	even	8	1
288.3.u.b		64	12.b	even	2	1
288.3.u.b		64	96.p	odd	8	1
384.3.m.a		64	1.a	even	1	1	trivial
384.3.m.a		64	32.h	odd	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(384, [\chi])\).