LMFDB - Newform orbit 800.2.o.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(143,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("800.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	800.o (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:	$6.38803216170$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
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 200)
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (2 i - 2) q^{3} - 5 i q^{9} + 6 q^{11} + (4 i + 4) q^{17} + 2 i q^{19} + (4 i + 4) q^{27} + (12 i - 12) q^{33} - 6 q^{41} + (6 i - 6) q^{43} + 7 i q^{49} - 16 q^{51} + ( - 4 i - 4) q^{57} + 6 i q^{59} + \cdots - 30 i q^{99} +O(q^{100})$ q + (2i - 2) q^3 - 5i q^9 + 6 * q^11 + (4i + 4) q^17 + 2i q^19 + (4i + 4) q^27 + (12i - 12) q^33 - 6 * q^41 + (6i - 6) q^43 + 7i q^49 - 16 * q^51 + (-4i - 4) q^57 + 6i q^59 + (6i + 6) q^67 + (-12i + 12) q^73 - q^81 + (2i - 2) q^83 + 18i q^89 + (12i + 12) q^97 - 30i q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{3} + 12 q^{11} + 8 q^{17} + 8 q^{27} - 24 q^{33} - 12 q^{41} - 12 q^{43} - 32 q^{51} - 8 q^{57} + 12 q^{67} + 24 q^{73} - 2 q^{81} - 4 q^{83} + 24 q^{97}+O(q^{100})$ 2 * q - 4 * q^3 + 12 * q^11 + 8 * q^17 + 8 * q^27 - 24 * q^33 - 12 * q^41 - 12 * q^43 - 32 * q^51 - 8 * q^57 + 12 * q^67 + 24 * q^73 - 2 * q^81 - 4 * q^83 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$ .

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$i$

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}

143.1

	−	1.00000i
		1.00000i

−2.00000

−

2.00000i

5.00000i

207.1

−2.00000

2.00000i

−

5.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2})$
5.c	odd	4	1	inner
40.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	800.2.o.a		2
4.b	odd	2	1		200.2.k.d	yes	2
5.b	even	2	1		800.2.o.d		2
5.c	odd	4	1	inner	800.2.o.a		2
5.c	odd	4	1		800.2.o.d		2
8.b	even	2	1		200.2.k.d	yes	2
8.d	odd	2	1	CM	800.2.o.a		2
20.d	odd	2	1		200.2.k.a	✓	2
20.e	even	4	1		200.2.k.a	✓	2
20.e	even	4	1		200.2.k.d	yes	2
40.e	odd	2	1		800.2.o.d		2
40.f	even	2	1		200.2.k.a	✓	2
40.i	odd	4	1		200.2.k.a	✓	2
40.i	odd	4	1		200.2.k.d	yes	2
40.k	even	4	1	inner	800.2.o.a		2
40.k	even	4	1		800.2.o.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.k.a	✓	2	20.d	odd	2	1
200.2.k.a	✓	2	20.e	even	4	1
200.2.k.a	✓	2	40.f	even	2	1
200.2.k.a	✓	2	40.i	odd	4	1
200.2.k.d	yes	2	4.b	odd	2	1
200.2.k.d	yes	2	8.b	even	2	1
200.2.k.d	yes	2	20.e	even	4	1
200.2.k.d	yes	2	40.i	odd	4	1
800.2.o.a		2	1.a	even	1	1	trivial
800.2.o.a		2	5.c	odd	4	1	inner
800.2.o.a		2	8.d	odd	2	1	CM
800.2.o.a		2	40.k	even	4	1	inner
800.2.o.d		2	5.b	even	2	1
800.2.o.d		2	5.c	odd	4	1
800.2.o.d		2	40.e	odd	2	1
800.2.o.d		2	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(800, [\chi])$ :

T_{3}^{2} + 4T_{3} + 8

T_{7}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 4T + 8$ T^2 + 4*T + 8
$5$	$T^{2}$ T^2
$7$	$T^{2}$ T^2
$11$	$(T - 6)^{2}$ (T - 6)^2
$13$	$T^{2}$ T^2
$17$	$T^{2} - 8T + 32$ T^2 - 8*T + 32
$19$	$T^{2} + 4$ T^2 + 4
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2}$ T^2
$37$	$T^{2}$ T^2
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$T^{2} + 12T + 72$ T^2 + 12*T + 72
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2} + 36$ T^2 + 36
$61$	$T^{2}$ T^2
$67$	$T^{2} - 12T + 72$ T^2 - 12*T + 72
$71$	$T^{2}$ T^2
$73$	$T^{2} - 24T + 288$ T^2 - 24*T + 288
$79$	$T^{2}$ T^2
$83$	$T^{2} + 4T + 8$ T^2 + 4*T + 8
$89$	$T^{2} + 324$ T^2 + 324
$97$	$T^{2} - 24T + 288$ T^2 - 24*T + 288
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