Newform orbit 2800.1.f.b

• Label: 2800.1.f.b
• Level: $2800$
• Weight: $1$
• Character orbit: 2800.f
• Self dual: yes
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -7
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2800.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 175)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.175.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.9800000.1
Stark unit:	Root of $x^{6} - 47766x^{5} + 13800415x^{4} - 2254085460x^{3} + 13800415x^{2} - 47766x + 1$

$q$-expansion

\(f(q)\)

\(=\)

\( q + q^{7} + q^{9} + q^{11} - q^{23} - q^{29} + q^{37} - q^{43} + q^{49} - 2 q^{53} + q^{63} - q^{67} + q^{71} + q^{77} + q^{79} + q^{81} + q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times\).

\(n\)	\(351\)	\(801\)	\(2101\)	\(2577\)
\(\chi(n)\)	\(0\)	\(1\)	\(0\)	\(0\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2001.1

0

0

0

0

0

0

1.00000

0

1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2800.1.f.b		1
4.b	odd	2	1		175.1.d.b	yes	1
5.b	even	2	1		2800.1.f.a		1
5.c	odd	4	2		2800.1.p.a		2
7.b	odd	2	1	CM	2800.1.f.b		1
12.b	even	2	1		1575.1.h.a		1
20.d	odd	2	1		175.1.d.a	✓	1
20.e	even	4	2		175.1.c.a		2
28.d	even	2	1		175.1.d.b	yes	1
28.f	even	6	2		1225.1.i.a		2
28.g	odd	6	2		1225.1.i.a		2
35.c	odd	2	1		2800.1.f.a		1
35.f	even	4	2		2800.1.p.a		2
60.h	even	2	1		1575.1.h.c		1
60.l	odd	4	2		1575.1.e.a		2
84.h	odd	2	1		1575.1.h.a		1
140.c	even	2	1		175.1.d.a	✓	1
140.j	odd	4	2		175.1.c.a		2
140.p	odd	6	2		1225.1.i.b		2
140.s	even	6	2		1225.1.i.b		2
140.w	even	12	4		1225.1.j.a		4
140.x	odd	12	4		1225.1.j.a		4
420.o	odd	2	1		1575.1.h.c		1
420.w	even	4	2		1575.1.e.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
175.1.c.a		2	20.e	even	4	2
175.1.c.a		2	140.j	odd	4	2
175.1.d.a	✓	1	20.d	odd	2	1
175.1.d.a	✓	1	140.c	even	2	1
175.1.d.b	yes	1	4.b	odd	2	1
175.1.d.b	yes	1	28.d	even	2	1
1225.1.i.a		2	28.f	even	6	2
1225.1.i.a		2	28.g	odd	6	2
1225.1.i.b		2	140.p	odd	6	2
1225.1.i.b		2	140.s	even	6	2
1225.1.j.a		4	140.w	even	12	4
1225.1.j.a		4	140.x	odd	12	4
1575.1.e.a		2	60.l	odd	4	2
1575.1.e.a		2	420.w	even	4	2
1575.1.h.a		1	12.b	even	2	1
1575.1.h.a		1	84.h	odd	2	1
1575.1.h.c		1	60.h	even	2	1
1575.1.h.c		1	420.o	odd	2	1
2800.1.f.a		1	5.b	even	2	1
2800.1.f.a		1	35.c	odd	2	1
2800.1.f.b		1	1.a	even	1	1	trivial
2800.1.f.b		1	7.b	odd	2	1	CM
2800.1.p.a		2	5.c	odd	4	2
2800.1.p.a		2	35.f	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2800, [\chi])\):

\( T_{3} \)

\( T_{23} + 1 \)

Hecke characteristic polynomials

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