Newform orbit 800.5.e.e

• Label: 800.5.e.e
• Level: $800$
• Weight: $5$
• Character orbit: 800.e
• Analytic conductor: $82.696$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	800.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(82.6959704671\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
399.1		0			−	5.51460i		0			0			0		78.3513				0		50.5892				0
399.2		0				5.51460i		0			0			0		78.3513				0		50.5892				0
399.3		0			−	13.1346i		0			0			0		−61.6422				0		−91.5179				0
399.4		0				13.1346i		0			0			0		−61.6422				0		−91.5179				0
399.5		0			−	15.9375i		0			0			0		−56.7751				0		−173.005				0
399.6		0				15.9375i		0			0			0		−56.7751				0		−173.005				0
399.7		0			−	15.2196i		0			0			0		47.5956				0		−150.635				0
399.8		0				15.2196i		0			0			0		47.5956				0		−150.635				0
399.9		0			−	10.2034i		0			0			0		43.3025				0		−23.1094				0
399.10		0				10.2034i		0			0			0		43.3025				0		−23.1094				0
399.11		0			−	4.51805i		0			0			0		−50.0881				0		60.5872				0
399.12		0				4.51805i		0			0			0		−50.0881				0		60.5872				0
399.13		0			−	0.715641i		0			0			0		25.0983				0		80.4879				0
399.14		0				0.715641i		0			0			0		25.0983				0		80.4879				0
399.15		0			−	7.09911i		0			0			0		2.59084				0		30.6027				0
399.16		0				7.09911i		0			0			0		2.59084				0		30.6027				0
399.17		0			−	7.09911i		0			0			0		−2.59084				0		30.6027				0
399.18		0				7.09911i		0			0			0		−2.59084				0		30.6027				0
399.19		0			−	0.715641i		0			0			0		−25.0983				0		80.4879				0
399.20		0				0.715641i		0			0			0		−25.0983				0		80.4879				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.d	odd	2	1	inner
40.e	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	800.5.e.e		32
4.b	odd	2	1		200.5.e.e		32
5.b	even	2	1	inner	800.5.e.e		32
5.c	odd	4	1		160.5.g.a		16
5.c	odd	4	1		800.5.g.h		16
8.b	even	2	1		200.5.e.e		32
8.d	odd	2	1	inner	800.5.e.e		32
15.e	even	4	1		1440.5.g.a		16
20.d	odd	2	1		200.5.e.e		32
20.e	even	4	1		40.5.g.a	✓	16
20.e	even	4	1		200.5.g.h		16
40.e	odd	2	1	inner	800.5.e.e		32
40.f	even	2	1		200.5.e.e		32
40.i	odd	4	1		40.5.g.a	✓	16
40.i	odd	4	1		200.5.g.h		16
40.k	even	4	1		160.5.g.a		16
40.k	even	4	1		800.5.g.h		16
60.l	odd	4	1		360.5.g.a		16
120.q	odd	4	1		1440.5.g.a		16
120.w	even	4	1		360.5.g.a		16

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.5.g.a	✓	16	20.e	even	4	1
40.5.g.a	✓	16	40.i	odd	4	1
160.5.g.a		16	5.c	odd	4	1
160.5.g.a		16	40.k	even	4	1
200.5.e.e		32	4.b	odd	2	1
200.5.e.e		32	8.b	even	2	1
200.5.e.e		32	20.d	odd	2	1
200.5.e.e		32	40.f	even	2	1
200.5.g.h		16	20.e	even	4	1
200.5.g.h		16	40.i	odd	4	1
360.5.g.a		16	60.l	odd	4	1
360.5.g.a		16	120.w	even	4	1
800.5.e.e		32	1.a	even	1	1	trivial
800.5.e.e		32	5.b	even	2	1	inner
800.5.e.e		32	8.d	odd	2	1	inner
800.5.e.e		32	40.e	odd	2	1	inner
800.5.g.h		16	5.c	odd	4	1
800.5.g.h		16	40.k	even	4	1
1440.5.g.a		16	15.e	even	4	1
1440.5.g.a		16	120.q	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^16 + 864*T3^14 + 291920*T3^12 + 48976000*T3^10 + 4307809120*T3^8 + 195311264256*T3^6 + 4243684379904*T3^4 + 35183065651200*T3^2 + 16931579040000