LMFDB - Newform orbit 1600.1.p.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,1,Mod(193,1600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1600.193");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1600.p (of order $4$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.798504020213$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 160)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.2000.1
Artin image:	$C_4\wr C_2$
Artin field:	Galois closure of 8.0.32768000.1

$q$ -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.

$f(q)$	$=$	$q + i q^{9} + (i - 1) q^{13} + (i + 1) q^{17} + (i + 1) q^{37} - i q^{49} + ( - i + 1) q^{53} + (i - 1) q^{73} - q^{81} + ( - i - 1) q^{97} +O(q^{100})$ q + z * q^9 + (z - 1) * q^13 + (z + 1) * q^17 + (z + 1) * q^37 - z * q^49 + (-z + 1) * q^53 + (z - 1) * q^73 - q^81 + (-z - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{13} + 2 q^{17} + 2 q^{37} + 2 q^{53} - 2 q^{73} - 2 q^{81} - 2 q^{97}+O(q^{100})$ 2 * q - 2 * q^13 + 2 * q^17 + 2 * q^37 + 2 * q^53 - 2 * q^73 - 2 * q^81 - 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$ .

$n$	$577$	$901$	$1151$
$\chi(n)$	$i$	$1$	$1$

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}

193.1

	−	1.00000i
		1.00000i

−

1.00000i

257.1

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
5.c	odd	4	1	inner
20.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1600.1.p.b		2
4.b	odd	2	1	CM	1600.1.p.b		2
5.b	even	2	1		320.1.p.a		2
5.c	odd	4	1		320.1.p.a		2
5.c	odd	4	1	inner	1600.1.p.b		2
8.b	even	2	1		800.1.p.b		2
8.d	odd	2	1		800.1.p.b		2
15.d	odd	2	1		2880.1.bh.b		2
15.e	even	4	1		2880.1.bh.b		2
20.d	odd	2	1		320.1.p.a		2
20.e	even	4	1		320.1.p.a		2
20.e	even	4	1	inner	1600.1.p.b		2
40.e	odd	2	1		160.1.p.a	✓	2
40.f	even	2	1		160.1.p.a	✓	2
40.i	odd	4	1		160.1.p.a	✓	2
40.i	odd	4	1		800.1.p.b		2
40.k	even	4	1		160.1.p.a	✓	2
40.k	even	4	1		800.1.p.b		2
60.h	even	2	1		2880.1.bh.b		2
60.l	odd	4	1		2880.1.bh.b		2
80.i	odd	4	1		1280.1.m.b		2
80.j	even	4	1		1280.1.m.a		2
80.k	odd	4	1		1280.1.m.a		2
80.k	odd	4	1		1280.1.m.b		2
80.q	even	4	1		1280.1.m.a		2
80.q	even	4	1		1280.1.m.b		2
80.s	even	4	1		1280.1.m.b		2
80.t	odd	4	1		1280.1.m.a		2
120.i	odd	2	1		1440.1.bh.b		2
120.m	even	2	1		1440.1.bh.b		2
120.q	odd	4	1		1440.1.bh.b		2
120.w	even	4	1		1440.1.bh.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.1.p.a	✓	2	40.e	odd	2	1
160.1.p.a	✓	2	40.f	even	2	1
160.1.p.a	✓	2	40.i	odd	4	1
160.1.p.a	✓	2	40.k	even	4	1
320.1.p.a		2	5.b	even	2	1
320.1.p.a		2	5.c	odd	4	1
320.1.p.a		2	20.d	odd	2	1
320.1.p.a		2	20.e	even	4	1
800.1.p.b		2	8.b	even	2	1
800.1.p.b		2	8.d	odd	2	1
800.1.p.b		2	40.i	odd	4	1
800.1.p.b		2	40.k	even	4	1
1280.1.m.a		2	80.j	even	4	1
1280.1.m.a		2	80.k	odd	4	1
1280.1.m.a		2	80.q	even	4	1
1280.1.m.a		2	80.t	odd	4	1
1280.1.m.b		2	80.i	odd	4	1
1280.1.m.b		2	80.k	odd	4	1
1280.1.m.b		2	80.q	even	4	1
1280.1.m.b		2	80.s	even	4	1
1440.1.bh.b		2	120.i	odd	2	1
1440.1.bh.b		2	120.m	even	2	1
1440.1.bh.b		2	120.q	odd	4	1
1440.1.bh.b		2	120.w	even	4	1
1600.1.p.b		2	1.a	even	1	1	trivial
1600.1.p.b		2	4.b	odd	2	1	CM
1600.1.p.b		2	5.c	odd	4	1	inner
1600.1.p.b		2	20.e	even	4	1	inner
2880.1.bh.b		2	15.d	odd	2	1
2880.1.bh.b		2	15.e	even	4	1
2880.1.bh.b		2	60.h	even	2	1
2880.1.bh.b		2	60.l	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}$ T3 acting on $S_{1}^{\mathrm{new}}(1600, [\chi])$ .

Hecke characteristic polynomials

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