LMFDB - Newform orbit 2800.1.p.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2800,1,Mod(2449,2800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2800.2449"); S:= CuspForms(chi, 1); N := Newforms(S);

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

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.39738203537$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ 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

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

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

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

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

−

1.00000i

−1.00000

2449.2

1.00000i

−1.00000

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

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} + 1$ T^2 + 1
$11$	$(T - 1)^{2}$ (T - 1)^2
$13$	$T^{2}$ T^2
$17$	$T^{2}$ T^2
$19$	$T^{2}$ T^2
$23$	$T^{2} + 1$ T^2 + 1
$29$	$(T - 1)^{2}$ (T - 1)^2
$31$	$T^{2}$ T^2
$37$	$T^{2} + 1$ T^2 + 1
$41$	$T^{2}$ T^2
$43$	$T^{2} + 1$ T^2 + 1
$47$	$T^{2}$ T^2
$53$	$T^{2} + 4$ T^2 + 4
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$T^{2} + 1$ T^2 + 1
$71$	$(T - 1)^{2}$ (T - 1)^2
$73$	$T^{2}$ T^2
$79$	$(T + 1)^{2}$ (T + 1)^2
$83$	$T^{2}$ T^2
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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