LMFDB - Newform orbit 4788.2.w.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(505,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

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

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

chi := DirichletCharacter("4788.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4788.w (of order $3$ , degree $2$ , 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:	$38.2323724878$
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:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 532)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (3 \zeta_{6} - 3) q^{5} + q^{7} - 5 \zeta_{6} q^{13} + (3 \zeta_{6} - 3) q^{17} + ( - 2 \zeta_{6} + 5) q^{19} - 3 \zeta_{6} q^{23} - 4 \zeta_{6} q^{25} + 3 \zeta_{6} q^{29} + 8 q^{31} + (3 \zeta_{6} - 3) q^{35} + \cdots + (17 \zeta_{6} - 17) q^{97} +O(q^{100})$ q + (3z - 3) q^5 + q^7 - 5z q^13 + (3z - 3) q^17 + (-2z + 5) q^19 - 3z q^23 - 4z q^25 + 3z q^29 + 8 * q^31 + (3z - 3) q^35 + 2 * q^37 + (3z - 3) q^41 + (-z + 1) * q^43 - 9z q^47 + q^49 + 3z q^53 + (-9z + 9) q^59 + 7z q^61 + 15 * q^65 - 5z q^67 + (-3z + 3) q^71 + (-7z + 7) q^73 + (-z + 1) * q^79 + 12 * q^83 - 9z q^85 + 9z q^89 - 5z q^91 + (15z - 9) q^95 + (17z - 17) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{5} + 2 q^{7} - 5 q^{13} - 3 q^{17} + 8 q^{19} - 3 q^{23} - 4 q^{25} + 3 q^{29} + 16 q^{31} - 3 q^{35} + 4 q^{37} - 3 q^{41} + q^{43} - 9 q^{47} + 2 q^{49} + 3 q^{53} + 9 q^{59} + 7 q^{61} + 30 q^{65}+ \cdots - 17 q^{97}+O(q^{100})$ 2 * q - 3 * q^5 + 2 * q^7 - 5 * q^13 - 3 * q^17 + 8 * q^19 - 3 * q^23 - 4 * q^25 + 3 * q^29 + 16 * q^31 - 3 * q^35 + 4 * q^37 - 3 * q^41 + q^43 - 9 * q^47 + 2 * q^49 + 3 * q^53 + 9 * q^59 + 7 * q^61 + 30 * q^65 - 5 * q^67 + 3 * q^71 + 7 * q^73 + q^79 + 24 * q^83 - 9 * q^85 + 9 * q^89 - 5 * q^91 - 3 * q^95 - 17 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4788\mathbb{Z}\right)^\times$ .

$n$	$533$	$1009$	$2395$	$4105$
$\chi(n)$	$1$	$-\zeta_{6}$	$1$	$1$

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}

505.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.50000

2.59808i

1.00000

1261.1

−1.50000

−

2.59808i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4788.2.w.b		2
3.b	odd	2	1		532.2.j.a	✓	2
19.c	even	3	1	inner	4788.2.w.b		2
57.h	odd	6	1		532.2.j.a	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
532.2.j.a	✓	2	3.b	odd	2	1
532.2.j.a	✓	2	57.h	odd	6	1
4788.2.w.b		2	1.a	even	1	1	trivial
4788.2.w.b		2	19.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4788, [\chi])$ :

T_{5}^{2} + 3T_{5} + 9

T_{11}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$7$	$(T - 1)^{2}$ (T - 1)^2
$11$	$T^{2}$ T^2
$13$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$17$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$19$	$T^{2} - 8T + 19$ T^2 - 8*T + 19
$23$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$29$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$31$	$(T - 8)^{2}$ (T - 8)^2
$37$	$(T - 2)^{2}$ (T - 2)^2
$41$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$43$	$T^{2} - T + 1$ T^2 - T + 1
$47$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$53$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$59$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$61$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$67$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$71$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$73$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$79$	$T^{2} - T + 1$ T^2 - T + 1
$83$	$(T - 12)^{2}$ (T - 12)^2
$89$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$97$	$T^{2} + 17T + 289$ T^2 + 17*T + 289
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