Newform orbit 384.4.n.a

• Label: 384.4.n.a
• Level: $384$
• Weight: $4$
• Character orbit: 384.n
• Analytic conductor: $22.657$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(22.6567334422\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
49.1		0		−2.77164	+	1.14805i		0		−3.82395	+	9.23183i		0		0.902320	+	0.902320i		0		6.36396	−	6.36396i		0
49.2		0		−2.77164	+	1.14805i		0		−3.59912	+	8.68905i		0		−4.07596	−	4.07596i		0		6.36396	−	6.36396i		0
49.3		0		−2.77164	+	1.14805i		0		3.92474	−	9.47517i		0		9.05205	+	9.05205i		0		6.36396	−	6.36396i		0
49.4		0		−2.77164	+	1.14805i		0		−2.47683	+	5.97959i		0		−17.4348	−	17.4348i		0		6.36396	−	6.36396i		0
49.5		0		−2.77164	+	1.14805i		0		1.39870	−	3.37677i		0		−19.0279	−	19.0279i		0		6.36396	−	6.36396i		0
49.6		0		−2.77164	+	1.14805i		0		0.245889	−	0.593629i		0		22.6211	+	22.6211i		0		6.36396	−	6.36396i		0
49.7		0		−2.77164	+	1.14805i		0		0.637895	−	1.54002i		0		21.6359	+	21.6359i		0		6.36396	−	6.36396i		0
49.8		0		−2.77164	+	1.14805i		0		4.92948	−	11.9008i		0		−3.52308	−	3.52308i		0		6.36396	−	6.36396i		0
49.9		0		−2.77164	+	1.14805i		0		5.64591	−	13.6304i		0		−16.7100	−	16.7100i		0		6.36396	−	6.36396i		0
49.10		0		−2.77164	+	1.14805i		0		−5.33520	+	12.8803i		0		−5.00776	−	5.00776i		0		6.36396	−	6.36396i		0
49.11		0		−2.77164	+	1.14805i		0		−5.92903	+	14.3139i		0		16.3325	+	16.3325i		0		6.36396	−	6.36396i		0
49.12		0		−2.77164	+	1.14805i		0		8.20833	−	19.8167i		0		5.13502	+	5.13502i		0		6.36396	−	6.36396i		0
49.13		0		2.77164	−	1.14805i		0		7.74531	−	18.6988i		0		−17.6550	−	17.6550i		0		6.36396	−	6.36396i		0
49.14		0		2.77164	−	1.14805i		0		5.41058	−	13.0623i		0		20.8641	+	20.8641i		0		6.36396	−	6.36396i		0
49.15		0		2.77164	−	1.14805i		0		3.34846	−	8.08390i		0		−8.76230	−	8.76230i		0		6.36396	−	6.36396i		0
49.16		0		2.77164	−	1.14805i		0		−2.19526	+	5.29982i		0		10.7161	+	10.7161i		0		6.36396	−	6.36396i		0
49.17		0		2.77164	−	1.14805i		0		−1.90077	+	4.58887i		0		−10.6034	−	10.6034i		0		6.36396	−	6.36396i		0
49.18		0		2.77164	−	1.14805i		0		1.12181	−	2.70830i		0		2.57989	+	2.57989i		0		6.36396	−	6.36396i		0
49.19		0		2.77164	−	1.14805i		0		0.866110	−	2.09098i		0		11.6959	+	11.6959i		0		6.36396	−	6.36396i		0
49.20		0		2.77164	−	1.14805i		0		5.49171	−	13.2582i		0		1.23403	+	1.23403i		0		6.36396	−	6.36396i		0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	384.4.n.a		96
4.b	odd	2	1		96.4.n.a	✓	96
32.g	even	8	1	inner	384.4.n.a		96
32.h	odd	8	1		96.4.n.a	✓	96

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.4.n.a	✓	96	4.b	odd	2	1
96.4.n.a	✓	96	32.h	odd	8	1
384.4.n.a		96	1.a	even	1	1	trivial
384.4.n.a		96	32.g	even	8	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(384, [\chi])\).