Newform orbit 2800.1.p.a

• Label: 2800.1.p.a
• Level: $2800$
• Weight: $1$
• Character orbit: 2800.p
• Analytic conductor: $1.397$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{3}$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.39738203537\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
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

$q$-expansion

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

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

Character values

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

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

2449.1

		1.00000i
	−	1.00000i

0

0

0

0

0

−

1.00000i

0

−1.00000

0

2449.2

0

0

0

0

0

1.00000i

0

−1.00000

0

Inner twists

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

Twists

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

    

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

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2800, [\chi])\).

Hecke characteristic polynomials

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