Newform orbit 2200.1.w.b

• Label: 2200.1.w.b
• Level: $2200$
• Weight: $1$
• Character orbit: 2200.w
• Analytic conductor: $1.098$
• Analytic rank: $0$
• Dimension: $4$
• Projective image: $D_{2}$
• CM/RM discs: -8, -55, 440
• Inner twists: $16$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2200.w (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.09794302779\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(i)\)
Coefficient field:	\(\Q(\zeta_{8})\)
Defining polynomial:	\( x^{4} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-2}, \sqrt{-55})\)
Artin image:	$\OD_{16}:C_2$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

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

\(f(q)\)	\(=\)	\( q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} - \zeta_{8} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\)	\(177\)	\(551\)	\(1101\)	\(1201\)
\(\chi(n)\)	\(\zeta_{8}^{2}\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

43.1

−0.707107	+	0.707107i
0.707107	−	0.707107i
−0.707107	−	0.707107i
0.707107	+	0.707107i

−0.707107

−

0.707107i

0

1.00000i

0

0

0

0.707107

−

0.707107i

−

1.00000i

0

43.2

0.707107

+

0.707107i

0

1.00000i

0

0

0

−0.707107

+

0.707107i

−

1.00000i

0

307.1

−0.707107

+

0.707107i

0

−

1.00000i

0

0

0

0.707107

+

0.707107i

1.00000i

0

307.2

0.707107

−

0.707107i

0

−

1.00000i

0

0

0

−0.707107

−

0.707107i

1.00000i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
55.d	odd	2	1	CM by \(\Q(\sqrt{-55}) \)
440.c	even	2	1	RM by \(\Q(\sqrt{110}) \)
5.b	even	2	1	inner
5.c	odd	4	2	inner
11.b	odd	2	1	inner
40.e	odd	2	1	inner
40.k	even	4	2	inner
55.e	even	4	2	inner
88.g	even	2	1	inner
440.w	odd	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2200.1.w.b	✓	4
5.b	even	2	1	inner	2200.1.w.b	✓	4
5.c	odd	4	2	inner	2200.1.w.b	✓	4
8.d	odd	2	1	CM	2200.1.w.b	✓	4
11.b	odd	2	1	inner	2200.1.w.b	✓	4
40.e	odd	2	1	inner	2200.1.w.b	✓	4
40.k	even	4	2	inner	2200.1.w.b	✓	4
55.d	odd	2	1	CM	2200.1.w.b	✓	4
55.e	even	4	2	inner	2200.1.w.b	✓	4
88.g	even	2	1	inner	2200.1.w.b	✓	4
440.c	even	2	1	RM	2200.1.w.b	✓	4
440.w	odd	4	2	inner	2200.1.w.b	✓	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2200.1.w.b	✓	4	1.a	even	1	1	trivial
2200.1.w.b	✓	4	5.b	even	2	1	inner
2200.1.w.b	✓	4	5.c	odd	4	2	inner
2200.1.w.b	✓	4	8.d	odd	2	1	CM
2200.1.w.b	✓	4	11.b	odd	2	1	inner
2200.1.w.b	✓	4	40.e	odd	2	1	inner
2200.1.w.b	✓	4	40.k	even	4	2	inner
2200.1.w.b	✓	4	55.d	odd	2	1	CM
2200.1.w.b	✓	4	55.e	even	4	2	inner
2200.1.w.b	✓	4	88.g	even	2	1	inner
2200.1.w.b	✓	4	440.c	even	2	1	RM
2200.1.w.b	✓	4	440.w	odd	4	2	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2200, [\chi])\):

\( T_{3} \)

\( T_{7} \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + 1 \)
$3$	\( T^{4} \)
$5$	\( T^{4} \)
$7$	\( T^{4} \)
$11$	\( (T - 1)^{4} \)