LMFDB - Newform orbit 2520.2.bi.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(361,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("2520.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2520.bi (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:	$20.1223013094$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 840)
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.

$f(q)$	$=$	$q - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 2) q^{7} + (2 \zeta_{6} - 2) q^{11} + q^{13} - 3 \zeta_{6} q^{19} + (\zeta_{6} - 1) q^{25} + ( - 9 \zeta_{6} + 9) q^{31} + (\zeta_{6} - 3) q^{35} - 3 \zeta_{6} q^{37} + \cdots - 14 q^{97} +O(q^{100})$ q - z * q^5 + (-3z + 2) q^7 + (2z - 2) q^11 + q^13 - 3z q^19 + (z - 1) * q^25 + (-9z + 9) q^31 + (z - 3) * q^35 - 3z q^37 - 2 * q^41 + 3 * q^43 - 6z q^47 + (-3z - 5) q^49 + 2 * q^55 + (4z - 4) q^59 - 2z q^61 - z * q^65 + (5z - 5) q^67 - 14 * q^71 + (z - 1) * q^73 + (4z + 2) q^77 - 9z q^79 + 6 * q^83 - 4z q^89 + (-3z + 2) q^91 + (3z - 3) q^95 - 14 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{5} + q^{7} - 2 q^{11} + 2 q^{13} - 3 q^{19} - q^{25} + 9 q^{31} - 5 q^{35} - 3 q^{37} - 4 q^{41} + 6 q^{43} - 6 q^{47} - 13 q^{49} + 4 q^{55} - 4 q^{59} - 2 q^{61} - q^{65} - 5 q^{67} - 28 q^{71}+ \cdots - 28 q^{97}+O(q^{100})$ 2 * q - q^5 + q^7 - 2 * q^11 + 2 * q^13 - 3 * q^19 - q^25 + 9 * q^31 - 5 * q^35 - 3 * q^37 - 4 * q^41 + 6 * q^43 - 6 * q^47 - 13 * q^49 + 4 * q^55 - 4 * q^59 - 2 * q^61 - q^65 - 5 * q^67 - 28 * q^71 - q^73 + 8 * q^77 - 9 * q^79 + 12 * q^83 - 4 * q^89 + q^91 - 3 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$ .

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$1$	$1$	$-\zeta_{6}$	$1$	$1$

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}

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

0.500000

−

2.59808i

1801.1

−0.500000

0.866025i

0.500000

2.59808i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2520.2.bi.c		2
3.b	odd	2	1		840.2.bg.e	✓	2
7.c	even	3	1	inner	2520.2.bi.c		2
12.b	even	2	1		1680.2.bg.h		2
21.g	even	6	1		5880.2.a.bc		1
21.h	odd	6	1		840.2.bg.e	✓	2
21.h	odd	6	1		5880.2.a.c		1
84.n	even	6	1		1680.2.bg.h		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
840.2.bg.e	✓	2	3.b	odd	2	1
840.2.bg.e	✓	2	21.h	odd	6	1
1680.2.bg.h		2	12.b	even	2	1
1680.2.bg.h		2	84.n	even	6	1
2520.2.bi.c		2	1.a	even	1	1	trivial
2520.2.bi.c		2	7.c	even	3	1	inner
5880.2.a.c		1	21.h	odd	6	1
5880.2.a.bc		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2520, [\chi])$ :

T_{11}^{2} + 2T_{11} + 4

T_{13} - 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + T + 1$ T^2 + T + 1
$7$	$T^{2} - T + 7$ T^2 - T + 7
$11$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$13$	$(T - 1)^{2}$ (T - 1)^2
$17$	$T^{2}$ T^2
$19$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$23$	$T^{2}$ T^2
$29$	$T^{2}$ T^2
$31$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$37$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$41$	$(T + 2)^{2}$ (T + 2)^2
$43$	$(T - 3)^{2}$ (T - 3)^2
$47$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$53$	$T^{2}$ T^2
$59$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$61$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$67$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$71$	$(T + 14)^{2}$ (T + 14)^2
$73$	$T^{2} + T + 1$ T^2 + T + 1
$79$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$83$	$(T - 6)^{2}$ (T - 6)^2
$89$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$97$	$(T + 14)^{2}$ (T + 14)^2
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