Newform orbit 320.3.r.a

• Label: 320.3.r.a
• Level: $320$
• Weight: $3$
• Character orbit: 320.r
• Analytic conductor: $8.719$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.r (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.71936845953\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
111.1		0		−3.81615	+	3.81615i		0		1.58114	−	1.58114i		0		7.06228				0			−	20.1260i		0
111.2		0		−3.58499	+	3.58499i		0		−1.58114	+	1.58114i		0		−10.0090				0			−	16.7043i		0
111.3		0		−3.09850	+	3.09850i		0		−1.58114	+	1.58114i		0		13.0357				0			−	10.2014i		0
111.4		0		−2.58103	+	2.58103i		0		1.58114	−	1.58114i		0		−0.523915				0			−	4.32341i		0
111.5		0		−1.92589	+	1.92589i		0		1.58114	−	1.58114i		0		−4.10918				0				1.58188i		0
111.6		0		−1.91324	+	1.91324i		0		−1.58114	+	1.58114i		0		−1.82022				0				1.67900i		0
111.7		0		−0.0991349	+	0.0991349i		0		1.58114	−	1.58114i		0		3.75219				0				8.98034i		0
111.8		0		0.313290	−	0.313290i		0		1.58114	−	1.58114i		0		−10.1627				0				8.80370i		0
111.9		0		0.374900	−	0.374900i		0		−1.58114	+	1.58114i		0		−2.42442				0				8.71890i		0
111.10		0		1.39844	−	1.39844i		0		−1.58114	+	1.58114i		0		−9.44746				0				5.08871i		0
111.11		0		1.40411	−	1.40411i		0		−1.58114	+	1.58114i		0		−0.552025				0				5.05693i		0
111.12		0		1.45771	−	1.45771i		0		−1.58114	+	1.58114i		0		11.3889				0				4.75014i		0
111.13		0		2.05924	−	2.05924i		0		1.58114	−	1.58114i		0		−10.3931				0				0.519079i		0
111.14		0		2.56741	−	2.56741i		0		1.58114	−	1.58114i		0		8.10325				0			−	4.18314i		0
111.15		0		3.48227	−	3.48227i		0		1.58114	−	1.58114i		0		6.27123				0			−	15.2524i		0
111.16		0		3.96156	−	3.96156i		0		−1.58114	+	1.58114i		0		−0.171519				0			−	22.3880i		0
271.1		0		−3.81615	−	3.81615i		0		1.58114	+	1.58114i		0		7.06228				0				20.1260i		0
271.2		0		−3.58499	−	3.58499i		0		−1.58114	−	1.58114i		0		−10.0090				0				16.7043i		0
271.3		0		−3.09850	−	3.09850i		0		−1.58114	−	1.58114i		0		13.0357				0				10.2014i		0
271.4		0		−2.58103	−	2.58103i		0		1.58114	+	1.58114i		0		−0.523915				0				4.32341i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.3.r.a		32
4.b	odd	2	1		80.3.r.a	✓	32
8.b	even	2	1		640.3.r.b		32
8.d	odd	2	1		640.3.r.a		32
16.e	even	4	1		80.3.r.a	✓	32
16.e	even	4	1		640.3.r.a		32
16.f	odd	4	1	inner	320.3.r.a		32
16.f	odd	4	1		640.3.r.b		32
20.d	odd	2	1		400.3.r.f		32
20.e	even	4	1		400.3.k.g		32
20.e	even	4	1		400.3.k.h		32
80.i	odd	4	1		400.3.k.h		32
80.q	even	4	1		400.3.r.f		32
80.t	odd	4	1		400.3.k.g		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.3.r.a	✓	32	4.b	odd	2	1
80.3.r.a	✓	32	16.e	even	4	1
320.3.r.a		32	1.a	even	1	1	trivial
320.3.r.a		32	16.f	odd	4	1	inner
400.3.k.g		32	20.e	even	4	1
400.3.k.g		32	80.t	odd	4	1
400.3.k.h		32	20.e	even	4	1
400.3.k.h		32	80.i	odd	4	1
400.3.r.f		32	20.d	odd	2	1
400.3.r.f		32	80.q	even	4	1
640.3.r.a		32	8.d	odd	2	1
640.3.r.a		32	16.e	even	4	1
640.3.r.b		32	8.b	even	2	1
640.3.r.b		32	16.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(320, [\chi])\).