LMFDB - Newform orbit 5733.2.a.t

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5733 = 3^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5733.a (trivial)

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:	yes
Analytic conductor:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $\beta = \sqrt{3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{2} + q^{4} - \beta q^{8} - 2 \beta q^{11} - q^{13} - 5 q^{16} + 4 \beta q^{17} - 2 q^{19} - 6 q^{22} + 4 \beta q^{23} - 5 q^{25} - \beta q^{26} + 4 \beta q^{29} - 2 q^{31} - 3 \beta q^{32} + \cdots + 10 q^{97} +O(q^{100})$ q + b * q^2 + q^4 - b * q^8 - 2b q^11 - q^13 - 5 * q^16 + 4b q^17 - 2 * q^19 - 6 * q^22 + 4b q^23 - 5 * q^25 - b * q^26 + 4b q^29 - 2 * q^31 - 3b q^32 + 12 * q^34 + 2 * q^37 - 2b q^38 - 4b q^41 + 8 * q^43 - 2b q^44 + 12 * q^46 + 6b q^47 - 5b q^50 - q^52 + 12 * q^58 - 2b q^59 + 10 * q^61 - 2b q^62 + q^64 + 14 * q^67 + 4b q^68 + 2b q^71 + 10 * q^73 + 2b q^74 - 2 * q^76 - 4 * q^79 - 12 * q^82 - 6b q^83 + 8b q^86 + 6 * q^88 - 4b q^89 + 4b q^92 + 18 * q^94 + 10 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{4} - 2 q^{13} - 10 q^{16} - 4 q^{19} - 12 q^{22} - 10 q^{25} - 4 q^{31} + 24 q^{34} + 4 q^{37} + 16 q^{43} + 24 q^{46} - 2 q^{52} + 24 q^{58} + 20 q^{61} + 2 q^{64} + 28 q^{67} + 20 q^{73} - 4 q^{76}+ \cdots + 20 q^{97}+O(q^{100})$ 2 * q + 2 * q^4 - 2 * q^13 - 10 * q^16 - 4 * q^19 - 12 * q^22 - 10 * q^25 - 4 * q^31 + 24 * q^34 + 4 * q^37 + 16 * q^43 + 24 * q^46 - 2 * q^52 + 24 * q^58 + 20 * q^61 + 2 * q^64 + 28 * q^67 + 20 * q^73 - 4 * q^76 - 8 * q^79 - 24 * q^82 + 12 * q^88 + 36 * q^94 + 20 * q^97

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}

1.1

−1.73205
1.73205

−1.73205

1.00000

1.73205

1.2

1.73205

1.00000

−1.73205

Atkin-Lehner signs

$p$	Sign
$3$	$+1$
$7$	$-1$
$13$	$+1$

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5733.2.a.t		2
3.b	odd	2	1	inner	5733.2.a.t		2
7.b	odd	2	1		117.2.a.b	✓	2
21.c	even	2	1		117.2.a.b	✓	2
28.d	even	2	1		1872.2.a.v		2
35.c	odd	2	1		2925.2.a.y		2
35.f	even	4	2		2925.2.c.s		4
56.e	even	2	1		7488.2.a.cj		2
56.h	odd	2	1		7488.2.a.cq		2
63.l	odd	6	2		1053.2.e.i		4
63.o	even	6	2		1053.2.e.i		4
84.h	odd	2	1		1872.2.a.v		2
91.b	odd	2	1		1521.2.a.j		2
91.i	even	4	2		1521.2.b.i		4
105.g	even	2	1		2925.2.a.y		2
105.k	odd	4	2		2925.2.c.s		4
168.e	odd	2	1		7488.2.a.cj		2
168.i	even	2	1		7488.2.a.cq		2
273.g	even	2	1		1521.2.a.j		2
273.o	odd	4	2		1521.2.b.i		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.a.b	✓	2	7.b	odd	2	1
117.2.a.b	✓	2	21.c	even	2	1
1053.2.e.i		4	63.l	odd	6	2
1053.2.e.i		4	63.o	even	6	2
1521.2.a.j		2	91.b	odd	2	1
1521.2.a.j		2	273.g	even	2	1
1521.2.b.i		4	91.i	even	4	2
1521.2.b.i		4	273.o	odd	4	2
1872.2.a.v		2	28.d	even	2	1
1872.2.a.v		2	84.h	odd	2	1
2925.2.a.y		2	35.c	odd	2	1
2925.2.a.y		2	105.g	even	2	1
2925.2.c.s		4	35.f	even	4	2
2925.2.c.s		4	105.k	odd	4	2
5733.2.a.t		2	1.a	even	1	1	trivial
5733.2.a.t		2	3.b	odd	2	1	inner
7488.2.a.cj		2	56.e	even	2	1
7488.2.a.cj		2	168.e	odd	2	1
7488.2.a.cq		2	56.h	odd	2	1
7488.2.a.cq		2	168.i	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5733))$ :

T_{2}^{2} - 3

T_{5}

T_{11}^{2} - 12

T_{17}^{2} - 48

T_{19} + 2

T_{31} + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 3$ T^2 - 3
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2}$ T^2
$11$	$T^{2} - 12$ T^2 - 12
$13$	$(T + 1)^{2}$ (T + 1)^2
$17$	$T^{2} - 48$ T^2 - 48
$19$	$(T + 2)^{2}$ (T + 2)^2
$23$	$T^{2} - 48$ T^2 - 48
$29$	$T^{2} - 48$ T^2 - 48
$31$	$(T + 2)^{2}$ (T + 2)^2
$37$	$(T - 2)^{2}$ (T - 2)^2
$41$	$T^{2} - 48$ T^2 - 48
$43$	$(T - 8)^{2}$ (T - 8)^2
$47$	$T^{2} - 108$ T^2 - 108
$53$	$T^{2}$ T^2
$59$	$T^{2} - 12$ T^2 - 12
$61$	$(T - 10)^{2}$ (T - 10)^2
$67$	$(T - 14)^{2}$ (T - 14)^2
$71$	$T^{2} - 12$ T^2 - 12
$73$	$(T - 10)^{2}$ (T - 10)^2
$79$	$(T + 4)^{2}$ (T + 4)^2
$83$	$T^{2} - 108$ T^2 - 108
$89$	$T^{2} - 48$ T^2 - 48
$97$	$(T - 10)^{2}$ (T - 10)^2
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