Newform orbit 800.2.a.e

• Label: 800.2.a.e
• Level: $800$
• Weight: $2$
• Character orbit: 800.a
• Self dual: yes
• Analytic conductor: $6.388$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(6.38803216170\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	\(1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 3 q^{9}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

0

0

0

0

0

0

0

0

−3.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(-1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	800.2.a.e		1
3.b	odd	2	1		7200.2.a.y		1
4.b	odd	2	1	CM	800.2.a.e		1
5.b	even	2	1		800.2.a.f		1
5.c	odd	4	2		160.2.c.a	✓	2
8.b	even	2	1		1600.2.a.m		1
8.d	odd	2	1		1600.2.a.m		1
12.b	even	2	1		7200.2.a.y		1
15.d	odd	2	1		7200.2.a.bb		1
15.e	even	4	2		1440.2.f.c		2
20.d	odd	2	1		800.2.a.f		1
20.e	even	4	2		160.2.c.a	✓	2
40.e	odd	2	1		1600.2.a.l		1
40.f	even	2	1		1600.2.a.l		1
40.i	odd	4	2		320.2.c.a		2
40.k	even	4	2		320.2.c.a		2
60.h	even	2	1		7200.2.a.bb		1
60.l	odd	4	2		1440.2.f.c		2
80.i	odd	4	2		1280.2.f.c		2
80.j	even	4	2		1280.2.f.d		2
80.s	even	4	2		1280.2.f.c		2
80.t	odd	4	2		1280.2.f.d		2
120.q	odd	4	2		2880.2.f.n		2
120.w	even	4	2		2880.2.f.n		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.c.a	✓	2	5.c	odd	4	2
160.2.c.a	✓	2	20.e	even	4	2
320.2.c.a		2	40.i	odd	4	2
320.2.c.a		2	40.k	even	4	2
800.2.a.e		1	1.a	even	1	1	trivial
800.2.a.e		1	4.b	odd	2	1	CM
800.2.a.f		1	5.b	even	2	1
800.2.a.f		1	20.d	odd	2	1
1280.2.f.c		2	80.i	odd	4	2
1280.2.f.c		2	80.s	even	4	2
1280.2.f.d		2	80.j	even	4	2
1280.2.f.d		2	80.t	odd	4	2
1440.2.f.c		2	15.e	even	4	2
1440.2.f.c		2	60.l	odd	4	2
1600.2.a.l		1	40.e	odd	2	1
1600.2.a.l		1	40.f	even	2	1
1600.2.a.m		1	8.b	even	2	1
1600.2.a.m		1	8.d	odd	2	1
2880.2.f.n		2	120.q	odd	4	2
2880.2.f.n		2	120.w	even	4	2
7200.2.a.y		1	3.b	odd	2	1
7200.2.a.y		1	12.b	even	2	1
7200.2.a.bb		1	15.d	odd	2	1
7200.2.a.bb		1	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(800))\):

\( T_{3} \)

\( T_{11} \)

\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T \)
$11$	\( T \)