Newform orbit 200.1.e.a

• Label: 200.1.e.a
• Level: $200$
• Weight: $1$
• Character orbit: 200.e
• Analytic conductor: $0.100$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	200.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.0998130025266\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.200.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\)	\(=\)	\( q - i q^{2} - i q^{3} - q^{4} - q^{6} + i q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6}+O(q^{10}) \)

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

99.1

		1.00000i
	−	1.00000i

−

1.00000i

−

1.00000i

−1.00000

0

−1.00000

0

1.00000i

0

0

99.2

1.00000i

1.00000i

−1.00000

0

−1.00000

0

−

1.00000i

0

0

Inner twists

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

Twists

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

    

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

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(200, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 1 \)
$3$	\( T^{2} + 1 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( (T + 1)^{2} \)