Newform orbit 1800.1.g.b

• Label: 1800.1.g.b
• Level: $1800$
• Weight: $1$
• Character orbit: 1800.g
• Self dual: yes
• Analytic conductor: $0.898$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1800.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.898317022739\)
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 6.0.5400000.2
Stark unit:	Root of $x^{6} - 341826x^{5} + 377875275x^{4} - 116089947380x^{3} + 377875275x^{2} - 341826x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{2} + q^{4} + q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\)	\(577\)	\(901\)	\(1001\)	\(1351\)
\(\chi(n)\)	\(0\)	\(1\)	\(0\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

451.1

0

1.00000

0

1.00000

0

0

0

1.00000

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

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.1.e.a		2	15.e	even	4	2
200.1.e.a		2	120.q	odd	4	2
200.1.g.a	✓	1	3.b	odd	2	1
200.1.g.a	✓	1	24.f	even	2	1
200.1.g.b	yes	1	15.d	odd	2	1
200.1.g.b	yes	1	120.m	even	2	1
800.1.e.a		2	60.l	odd	4	2
800.1.e.a		2	120.w	even	4	2
800.1.g.a		1	12.b	even	2	1
800.1.g.a		1	24.h	odd	2	1
800.1.g.b		1	60.h	even	2	1
800.1.g.b		1	120.i	odd	2	1
1800.1.g.a		1	5.b	even	2	1
1800.1.g.a		1	40.e	odd	2	1
1800.1.g.b		1	1.a	even	1	1	trivial
1800.1.g.b		1	8.d	odd	2	1	CM
1800.1.p.a		2	5.c	odd	4	2
1800.1.p.a		2	40.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

\( T_{11} - 1 \)

\( T_{17} + 1 \)

Hecke characteristic polynomials

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