Newform orbit 800.1.g.b

• Label: 800.1.g.b
• Level: $800$
• Weight: $1$
• Character orbit: 800.g
• Self dual: yes
• Analytic conductor: $0.399$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	yes
Analytic conductor:	$0.399252010106$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 200)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.200.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.640000.1
Stark unit:	Root of $x^{6} - 386x^{5} - 1565x^{4} - 152900x^{3} - 1565x^{2} - 386x + 1$

$q$-expansion

$f(q)$

$=$

$q + q^{3} + q^{11} - q^{17} + q^{19} - q^{27} + q^{33} - q^{41} - 2 q^{43} + q^{49} - q^{51} + q^{57} - 2 q^{59} + q^{67} - q^{73} - q^{81} + q^{83} - q^{89} + 2 q^{97}+O(q^{100})$

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

1.00000

0

0

0

0

0

0

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	800.1.g.b		1
4.b	odd	2	1		200.1.g.b	yes	1
5.b	even	2	1		800.1.g.a		1
5.c	odd	4	2		800.1.e.a		2
8.b	even	2	1		200.1.g.b	yes	1
8.d	odd	2	1	CM	800.1.g.b		1
12.b	even	2	1		1800.1.g.a		1
20.d	odd	2	1		200.1.g.a	✓	1
20.e	even	4	2		200.1.e.a		2
24.h	odd	2	1		1800.1.g.a		1
40.e	odd	2	1		800.1.g.a		1
40.f	even	2	1		200.1.g.a	✓	1
40.i	odd	4	2		200.1.e.a		2
40.k	even	4	2		800.1.e.a		2
60.h	even	2	1		1800.1.g.b		1
60.l	odd	4	2		1800.1.p.a		2
120.i	odd	2	1		1800.1.g.b		1
120.w	even	4	2		1800.1.p.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.1.e.a		2	20.e	even	4	2
200.1.e.a		2	40.i	odd	4	2
200.1.g.a	✓	1	20.d	odd	2	1
200.1.g.a	✓	1	40.f	even	2	1
200.1.g.b	yes	1	4.b	odd	2	1
200.1.g.b	yes	1	8.b	even	2	1
800.1.e.a		2	5.c	odd	4	2
800.1.e.a		2	40.k	even	4	2
800.1.g.a		1	5.b	even	2	1
800.1.g.a		1	40.e	odd	2	1
800.1.g.b		1	1.a	even	1	1	trivial
800.1.g.b		1	8.d	odd	2	1	CM
1800.1.g.a		1	12.b	even	2	1
1800.1.g.a		1	24.h	odd	2	1
1800.1.g.b		1	60.h	even	2	1
1800.1.g.b		1	120.i	odd	2	1
1800.1.p.a		2	60.l	odd	4	2
1800.1.p.a		2	120.w	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 1$ acting on $S_{1}^{\mathrm{new}}(800, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$
$3$	$T - 1$
$5$	$T$
$7$	$T$
$11$	$T - 1$