LMFDB - Newform orbit 9360.2.a.cu

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{73})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 18$ x^2 - x - 18
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 780)
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 $\beta = \frac{1}{2}(1 + \sqrt{73})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + q^{5} + \beta q^{7} + \beta q^{11} - q^{13} + \beta q^{17} - 6 q^{19} + (\beta + 2) q^{23} + q^{25} - 2 q^{29} - 6 q^{31} + \beta q^{35} + \beta q^{37} + (\beta + 4) q^{41} - 8 q^{43} + 6 q^{47}+ \cdots + (3 \beta - 4) q^{97}+O(q^{100})$ q + q^5 + b * q^7 + b * q^11 - q^13 + b * q^17 - 6 * q^19 + (b + 2) * q^23 + q^25 - 2 * q^29 - 6 * q^31 + b * q^35 + b * q^37 + (b + 4) * q^41 - 8 * q^43 + 6 * q^47 + (b + 11) * q^49 + b * q^53 + b * q^55 + (-2b + 6) q^59 - b * q^61 - q^65 + (-2b + 2) q^67 + (-3b + 4) q^71 + 10 * q^73 + (b + 18) * q^77 + (b - 6) * q^79 + (2b - 2) q^83 + b * q^85 - b * q^89 - b * q^91 - 6 * q^95 + (3b - 4) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + q^{7} + q^{11} - 2 q^{13} + q^{17} - 12 q^{19} + 5 q^{23} + 2 q^{25} - 4 q^{29} - 12 q^{31} + q^{35} + q^{37} + 9 q^{41} - 16 q^{43} + 12 q^{47} + 23 q^{49} + q^{53} + q^{55} + 10 q^{59}+ \cdots - 5 q^{97}+O(q^{100})$ 2 * q + 2 * q^5 + q^7 + q^11 - 2 * q^13 + q^17 - 12 * q^19 + 5 * q^23 + 2 * q^25 - 4 * q^29 - 12 * q^31 + q^35 + q^37 + 9 * q^41 - 16 * q^43 + 12 * q^47 + 23 * q^49 + q^53 + q^55 + 10 * q^59 - q^61 - 2 * q^65 + 2 * q^67 + 5 * q^71 + 20 * q^73 + 37 * q^77 - 11 * q^79 - 2 * q^83 + q^85 - q^89 - q^91 - 12 * q^95 - 5 * 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

−3.77200
4.77200

1.00000

−3.77200

1.2

1.00000

4.77200

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$5$	$-1$
$13$	$+1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	9360.2.a.cu		2
3.b	odd	2	1		3120.2.a.be		2
4.b	odd	2	1		2340.2.a.l		2
12.b	even	2	1		780.2.a.e	✓	2
60.h	even	2	1		3900.2.a.s		2
60.l	odd	4	2		3900.2.h.i		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
780.2.a.e	✓	2	12.b	even	2	1
2340.2.a.l		2	4.b	odd	2	1
3120.2.a.be		2	3.b	odd	2	1
3900.2.a.s		2	60.h	even	2	1
3900.2.h.i		4	60.l	odd	4	2
9360.2.a.cu		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9360))$ :

T_{7}^{2} - T_{7} - 18

T_{11}^{2} - T_{11} - 18

T_{17}^{2} - T_{17} - 18

T_{19} + 6

T_{31} + 6

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$T^{2} - T - 18$ T^2 - T - 18
$11$	$T^{2} - T - 18$ T^2 - T - 18
$13$	$(T + 1)^{2}$ (T + 1)^2
$17$	$T^{2} - T - 18$ T^2 - T - 18
$19$	$(T + 6)^{2}$ (T + 6)^2
$23$	$T^{2} - 5T - 12$ T^2 - 5*T - 12
$29$	$(T + 2)^{2}$ (T + 2)^2
$31$	$(T + 6)^{2}$ (T + 6)^2
$37$	$T^{2} - T - 18$ T^2 - T - 18
$41$	$T^{2} - 9T + 2$ T^2 - 9*T + 2
$43$	$(T + 8)^{2}$ (T + 8)^2
$47$	$(T - 6)^{2}$ (T - 6)^2
$53$	$T^{2} - T - 18$ T^2 - T - 18
$59$	$T^{2} - 10T - 48$ T^2 - 10*T - 48
$61$	$T^{2} + T - 18$ T^2 + T - 18
$67$	$T^{2} - 2T - 72$ T^2 - 2*T - 72
$71$	$T^{2} - 5T - 158$ T^2 - 5*T - 158
$73$	$(T - 10)^{2}$ (T - 10)^2
$79$	$T^{2} + 11T + 12$ T^2 + 11*T + 12
$83$	$T^{2} + 2T - 72$ T^2 + 2*T - 72
$89$	$T^{2} + T - 18$ T^2 + T - 18
$97$	$T^{2} + 5T - 158$ T^2 + 5*T - 158
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more