LMFDB - Newform orbit 3240.2.q.w

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("3240.1081");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3240 = 2^{3} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$ , 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:	$25.8715302549$
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 1080)
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.

$f(q)$	$=$	$q + \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + (2 \zeta_{6} - 2) q^{11} - 4 \zeta_{6} q^{13} + q^{17} - 5 q^{19} - 5 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (8 \zeta_{6} - 8) q^{29} - 7 \zeta_{6} q^{31} + \cdots + ( - 8 \zeta_{6} + 8) q^{97} +O(q^{100})$ q + z * q^5 + (-4z + 4) q^7 + (2z - 2) q^11 - 4z q^13 + q^17 - 5 * q^19 - 5z q^23 + (z - 1) * q^25 + (8z - 8) q^29 - 7z q^31 + 4 * q^35 - 6 * q^37 - 6z q^41 + (-2z + 2) q^43 + (8z - 8) q^47 - 9z q^49 + 9 * q^53 - 2 * q^55 - 4z q^59 + (13z - 13) q^61 + (-4z + 4) q^65 + 10z q^67 - 6 * q^71 - 6 * q^73 + 8z q^77 + (9z - 9) q^79 + (-17z + 17) q^83 + z * q^85 - 6 * q^89 - 16 * q^91 - 5z q^95 + (-8z + 8) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{5} + 4 q^{7} - 2 q^{11} - 4 q^{13} + 2 q^{17} - 10 q^{19} - 5 q^{23} - q^{25} - 8 q^{29} - 7 q^{31} + 8 q^{35} - 12 q^{37} - 6 q^{41} + 2 q^{43} - 8 q^{47} - 9 q^{49} + 18 q^{53} - 4 q^{55} - 4 q^{59}+ \cdots + 8 q^{97}+O(q^{100})$ 2 * q + q^5 + 4 * q^7 - 2 * q^11 - 4 * q^13 + 2 * q^17 - 10 * q^19 - 5 * q^23 - q^25 - 8 * q^29 - 7 * q^31 + 8 * q^35 - 12 * q^37 - 6 * q^41 + 2 * q^43 - 8 * q^47 - 9 * q^49 + 18 * q^53 - 4 * q^55 - 4 * q^59 - 13 * q^61 + 4 * q^65 + 10 * q^67 - 12 * q^71 - 12 * q^73 + 8 * q^77 - 9 * q^79 + 17 * q^83 + q^85 - 12 * q^89 - 32 * q^91 - 5 * q^95 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$ .

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$-\zeta_{6}$

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}

1081.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0.500000

−

0.866025i

2.00000

3.46410i

2161.1

0.500000

0.866025i

2.00000

−

3.46410i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3240.2.q.w		2
3.b	odd	2	1		3240.2.q.l		2
9.c	even	3	1		1080.2.a.a	✓	1
9.c	even	3	1	inner	3240.2.q.w		2
9.d	odd	6	1		1080.2.a.g	yes	1
9.d	odd	6	1		3240.2.q.l		2
36.f	odd	6	1		2160.2.a.l		1
36.h	even	6	1		2160.2.a.w		1
45.h	odd	6	1		5400.2.a.br		1
45.j	even	6	1		5400.2.a.bu		1
45.k	odd	12	2		5400.2.f.s		2
45.l	even	12	2		5400.2.f.l		2
72.j	odd	6	1		8640.2.a.a		1
72.l	even	6	1		8640.2.a.bd		1
72.n	even	6	1		8640.2.a.bf		1
72.p	odd	6	1		8640.2.a.cg		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1080.2.a.a	✓	1	9.c	even	3	1
1080.2.a.g	yes	1	9.d	odd	6	1
2160.2.a.l		1	36.f	odd	6	1
2160.2.a.w		1	36.h	even	6	1
3240.2.q.l		2	3.b	odd	2	1
3240.2.q.l		2	9.d	odd	6	1
3240.2.q.w		2	1.a	even	1	1	trivial
3240.2.q.w		2	9.c	even	3	1	inner
5400.2.a.br		1	45.h	odd	6	1
5400.2.a.bu		1	45.j	even	6	1
5400.2.f.l		2	45.l	even	12	2
5400.2.f.s		2	45.k	odd	12	2
8640.2.a.a		1	72.j	odd	6	1
8640.2.a.bd		1	72.l	even	6	1
8640.2.a.bf		1	72.n	even	6	1
8640.2.a.cg		1	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3240, [\chi])$ :

T_{7}^{2} - 4T_{7} + 16

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

T_{17} - 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
$7$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$11$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$13$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$17$	$(T - 1)^{2}$ (T - 1)^2
$19$	$(T + 5)^{2}$ (T + 5)^2
$23$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$29$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$31$	$T^{2} + 7T + 49$ T^2 + 7*T + 49
$37$	$(T + 6)^{2}$ (T + 6)^2
$41$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$43$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$47$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$53$	$(T - 9)^{2}$ (T - 9)^2
$59$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$61$	$T^{2} + 13T + 169$ T^2 + 13*T + 169
$67$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$71$	$(T + 6)^{2}$ (T + 6)^2
$73$	$(T + 6)^{2}$ (T + 6)^2
$79$	$T^{2} + 9T + 81$ T^2 + 9*T + 81
$83$	$T^{2} - 17T + 289$ T^2 - 17*T + 289
$89$	$(T + 6)^{2}$ (T + 6)^2
$97$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
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