Newform orbit 200.1.g.a

• Label: 200.1.g.a
• Level: $200$
• Weight: $1$
• Character orbit: 200.g
• Self dual: yes
• Analytic conductor: $0.100$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -8
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	200.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.0998130025266\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.200.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.200000.1
Stark unit:	Root of $x^{6} - 11x^{5} + 40x^{4} - 65x^{3} + 40x^{2} - 11x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(151\)	\(177\)
\(\chi(n)\)	\(1\)	\(1\)	\(0\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

51.1

0

−1.00000

1.00000

1.00000

0

−1.00000

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

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.1.e.a		2	5.c	odd	4	2
200.1.e.a		2	40.k	even	4	2
200.1.g.a	✓	1	1.a	even	1	1	trivial
200.1.g.a	✓	1	8.d	odd	2	1	CM
200.1.g.b	yes	1	5.b	even	2	1
200.1.g.b	yes	1	40.e	odd	2	1
800.1.e.a		2	20.e	even	4	2
800.1.e.a		2	40.i	odd	4	2
800.1.g.a		1	4.b	odd	2	1
800.1.g.a		1	8.b	even	2	1
800.1.g.b		1	20.d	odd	2	1
800.1.g.b		1	40.f	even	2	1
1800.1.g.a		1	15.d	odd	2	1
1800.1.g.a		1	120.m	even	2	1
1800.1.g.b		1	3.b	odd	2	1
1800.1.g.b		1	24.f	even	2	1
1800.1.p.a		2	15.e	even	4	2
1800.1.p.a		2	120.q	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) acting on \(S_{1}^{\mathrm{new}}(200, [\chi])\).

Hecke characteristic polynomials

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