LMFDB - Newform orbit 3240.1.v.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,1,Mod(1297,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3240.1297");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3240 = 2^{3} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3240.v (of order $4$ , 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:	$1.61697064093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.0.162000.1

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q - q^{5} + q^{11} + i q^{19} + ( - i - 1) q^{23} + q^{25} + i q^{29} + q^{31} + ( - i + 1) q^{37} - q^{41} + (i + 1) q^{43} + ( - i + 1) q^{47} + i q^{49} - q^{55} + i q^{59} + ( - i + 1) q^{67}+ \cdots + (i - 1) q^{97}+O(q^{100})$ q - q^5 + q^11 + z * q^19 + (-z - 1) * q^23 + q^25 + z * q^29 + q^31 + (-z + 1) * q^37 - q^41 + (z + 1) * q^43 + (-z + 1) * q^47 + z * q^49 - q^55 + z * q^59 + (-z + 1) * q^67 + q^71 - 2z q^79 + (z + 1) * q^83 + z * q^89 - z * q^95 + (z - 1) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} + 2 q^{11} - 2 q^{23} + 2 q^{25} + 2 q^{31} + 2 q^{37} - 2 q^{41} + 2 q^{43} + 2 q^{47} - 2 q^{55} + 2 q^{67} + 2 q^{71} + 2 q^{83} - 2 q^{97}+O(q^{100})$ 2 * q - 2 * q^5 + 2 * q^11 - 2 * q^23 + 2 * q^25 + 2 * q^31 + 2 * q^37 - 2 * q^41 + 2 * q^43 + 2 * q^47 - 2 * q^55 + 2 * q^67 + 2 * q^71 + 2 * q^83 - 2 * q^97

Character values

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

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-i$	$1$	$1$	$1$

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}

1297.1

	−	1.00000i
		1.00000i

−1.00000

2593.1

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3240.1.v.a	✓	2
3.b	odd	2	1		3240.1.v.b	yes	2
5.c	odd	4	1	inner	3240.1.v.a	✓	2
9.c	even	3	2		3240.1.bq.b		4
9.d	odd	6	2		3240.1.bq.a		4
15.e	even	4	1		3240.1.v.b	yes	2
45.k	odd	12	2		3240.1.bq.b		4
45.l	even	12	2		3240.1.bq.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3240.1.v.a	✓	2	1.a	even	1	1	trivial
3240.1.v.a	✓	2	5.c	odd	4	1	inner
3240.1.v.b	yes	2	3.b	odd	2	1
3240.1.v.b	yes	2	15.e	even	4	1
3240.1.bq.a		4	9.d	odd	6	2
3240.1.bq.a		4	45.l	even	12	2
3240.1.bq.b		4	9.c	even	3	2
3240.1.bq.b		4	45.k	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{11} - 1$ T11 - 1 acting on $S_{1}^{\mathrm{new}}(3240, [\chi])$ .

Hecke characteristic polynomials

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