LMFDB - Newform orbit 800.2.d.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,2,Mod(401,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("800.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$800 = 2^{5} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	800.d (of order $2$ , degree $1$ , 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:	$\Q(\sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 2$ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $\beta = \sqrt{-7}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{3} + 4 q^{7} - 4 q^{9} - \beta q^{11} + 3 q^{17} - \beta q^{19} - 4 \beta q^{21} + 4 q^{23} + \beta q^{27} - 4 q^{31} - 7 q^{33} + 4 \beta q^{37} - 5 q^{41} - 2 \beta q^{43} - 8 q^{47} + 9 q^{49} + \cdots + 4 \beta q^{99} +O(q^{100})$ q - b * q^3 + 4 * q^7 - 4 * q^9 - b * q^11 + 3 * q^17 - b * q^19 - 4b q^21 + 4 * q^23 + b * q^27 - 4 * q^31 - 7 * q^33 + 4b q^37 - 5 * q^41 - 2b q^43 - 8 * q^47 + 9 * q^49 - 3b q^51 - 4b q^53 - 7 * q^57 - 2b q^59 + 4b q^61 - 16 * q^63 + 3b q^67 - 4b q^69 - 8 * q^71 + 7 * q^73 - 4b q^77 - 4 * q^79 - 5 * q^81 + 3b q^83 - q^89 + 4b q^93 + 2 * q^97 + 4b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 8 q^{7} - 8 q^{9} + 6 q^{17} + 8 q^{23} - 8 q^{31} - 14 q^{33} - 10 q^{41} - 16 q^{47} + 18 q^{49} - 14 q^{57} - 32 q^{63} - 16 q^{71} + 14 q^{73} - 8 q^{79} - 10 q^{81} - 2 q^{89} + 4 q^{97}+O(q^{100})$ 2 * q + 8 * q^7 - 8 * q^9 + 6 * q^17 + 8 * q^23 - 8 * q^31 - 14 * q^33 - 10 * q^41 - 16 * q^47 + 18 * q^49 - 14 * q^57 - 32 * q^63 - 16 * q^71 + 14 * q^73 - 8 * q^79 - 10 * q^81 - 2 * q^89 + 4 * q^97

Character values

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

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

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}

401.1

0.500000	+	1.32288i
0.500000	−	1.32288i

−

2.64575i

4.00000

−4.00000

401.2

2.64575i

4.00000

−4.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	800.2.d.d		2
3.b	odd	2	1		7200.2.k.i		2
4.b	odd	2	1		200.2.d.b	✓	2
5.b	even	2	1		800.2.d.a		2
5.c	odd	4	2		800.2.f.d		4
8.b	even	2	1	inner	800.2.d.d		2
8.d	odd	2	1		200.2.d.b	✓	2
12.b	even	2	1		1800.2.k.f		2
15.d	odd	2	1		7200.2.k.b		2
15.e	even	4	2		7200.2.d.m		4
16.e	even	4	2		6400.2.a.bh		2
16.f	odd	4	2		6400.2.a.cc		2
20.d	odd	2	1		200.2.d.c	yes	2
20.e	even	4	2		200.2.f.d		4
24.f	even	2	1		1800.2.k.f		2
24.h	odd	2	1		7200.2.k.i		2
40.e	odd	2	1		200.2.d.c	yes	2
40.f	even	2	1		800.2.d.a		2
40.i	odd	4	2		800.2.f.d		4
40.k	even	4	2		200.2.f.d		4
60.h	even	2	1		1800.2.k.d		2
60.l	odd	4	2		1800.2.d.m		4
80.k	odd	4	2		6400.2.a.bg		2
80.q	even	4	2		6400.2.a.cb		2
120.i	odd	2	1		7200.2.k.b		2
120.m	even	2	1		1800.2.k.d		2
120.q	odd	4	2		1800.2.d.m		4
120.w	even	4	2		7200.2.d.m		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.d.b	✓	2	4.b	odd	2	1
200.2.d.b	✓	2	8.d	odd	2	1
200.2.d.c	yes	2	20.d	odd	2	1
200.2.d.c	yes	2	40.e	odd	2	1
200.2.f.d		4	20.e	even	4	2
200.2.f.d		4	40.k	even	4	2
800.2.d.a		2	5.b	even	2	1
800.2.d.a		2	40.f	even	2	1
800.2.d.d		2	1.a	even	1	1	trivial
800.2.d.d		2	8.b	even	2	1	inner
800.2.f.d		4	5.c	odd	4	2
800.2.f.d		4	40.i	odd	4	2
1800.2.d.m		4	60.l	odd	4	2
1800.2.d.m		4	120.q	odd	4	2
1800.2.k.d		2	60.h	even	2	1
1800.2.k.d		2	120.m	even	2	1
1800.2.k.f		2	12.b	even	2	1
1800.2.k.f		2	24.f	even	2	1
6400.2.a.bg		2	80.k	odd	4	2
6400.2.a.bh		2	16.e	even	4	2
6400.2.a.cb		2	80.q	even	4	2
6400.2.a.cc		2	16.f	odd	4	2
7200.2.d.m		4	15.e	even	4	2
7200.2.d.m		4	120.w	even	4	2
7200.2.k.b		2	15.d	odd	2	1
7200.2.k.b		2	120.i	odd	2	1
7200.2.k.i		2	3.b	odd	2	1
7200.2.k.i		2	24.h	odd	2	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} + 7

T_{7} - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 7$ T^2 + 7
$5$	$T^{2}$ T^2
$7$	$(T - 4)^{2}$ (T - 4)^2
$11$	$T^{2} + 7$ T^2 + 7
$13$	$T^{2}$ T^2
$17$	$(T - 3)^{2}$ (T - 3)^2
$19$	$T^{2} + 7$ T^2 + 7
$23$	$(T - 4)^{2}$ (T - 4)^2
$29$	$T^{2}$ T^2
$31$	$(T + 4)^{2}$ (T + 4)^2
$37$	$T^{2} + 112$ T^2 + 112
$41$	$(T + 5)^{2}$ (T + 5)^2
$43$	$T^{2} + 28$ T^2 + 28
$47$	$(T + 8)^{2}$ (T + 8)^2
$53$	$T^{2} + 112$ T^2 + 112
$59$	$T^{2} + 28$ T^2 + 28
$61$	$T^{2} + 112$ T^2 + 112
$67$	$T^{2} + 63$ T^2 + 63
$71$	$(T + 8)^{2}$ (T + 8)^2
$73$	$(T - 7)^{2}$ (T - 7)^2
$79$	$(T + 4)^{2}$ (T + 4)^2
$83$	$T^{2} + 63$ T^2 + 63
$89$	$(T + 1)^{2}$ (T + 1)^2
$97$	$(T - 2)^{2}$ (T - 2)^2
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