Newform orbit 800.5.g.a

• Label: 800.5.g.a
• Level: $800$
• Weight: $5$
• Character orbit: 800.g
• Self dual: yes
• Analytic conductor: $82.696$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	\(82.6959704671\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\)

\(=\)

\( q - 14 q^{3} + 115 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(351\)	\(577\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

751.1

0

0

−14.0000

0

0

0

0

0

115.000

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	800.5.g.a		1
4.b	odd	2	1		200.5.g.a		1
5.b	even	2	1		32.5.d.a		1
5.c	odd	4	2		800.5.e.a		2
8.b	even	2	1		200.5.g.a		1
8.d	odd	2	1	CM	800.5.g.a		1
15.d	odd	2	1		288.5.b.a		1
20.d	odd	2	1		8.5.d.a	✓	1
20.e	even	4	2		200.5.e.a		2
40.e	odd	2	1		32.5.d.a		1
40.f	even	2	1		8.5.d.a	✓	1
40.i	odd	4	2		200.5.e.a		2
40.k	even	4	2		800.5.e.a		2
60.h	even	2	1		72.5.b.a		1
80.k	odd	4	2		256.5.c.d		2
80.q	even	4	2		256.5.c.d		2
120.i	odd	2	1		72.5.b.a		1
120.m	even	2	1		288.5.b.a		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8.5.d.a	✓	1	20.d	odd	2	1
8.5.d.a	✓	1	40.f	even	2	1
32.5.d.a		1	5.b	even	2	1
32.5.d.a		1	40.e	odd	2	1
72.5.b.a		1	60.h	even	2	1
72.5.b.a		1	120.i	odd	2	1
200.5.e.a		2	20.e	even	4	2
200.5.e.a		2	40.i	odd	4	2
200.5.g.a		1	4.b	odd	2	1
200.5.g.a		1	8.b	even	2	1
256.5.c.d		2	80.k	odd	4	2
256.5.c.d		2	80.q	even	4	2
288.5.b.a		1	15.d	odd	2	1
288.5.b.a		1	120.m	even	2	1
800.5.e.a		2	5.c	odd	4	2
800.5.e.a		2	40.k	even	4	2
800.5.g.a		1	1.a	even	1	1	trivial
800.5.g.a		1	8.d	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 14 \) acting on \(S_{5}^{\mathrm{new}}(800, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T \)
$3$	\( T + 14 \)
$5$	\( T \)
$7$	\( T \)
$11$	\( T - 46 \)