LMFDB - Newform orbit 9360.2.a.cm

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 65)
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 $\beta = \sqrt{3}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + q^{5} - 2 q^{7} + (\beta - 3) q^{11} + q^{13} - 2 \beta q^{17} + ( - 3 \beta + 1) q^{19} + (\beta + 3) q^{23} + q^{25} + (2 \beta + 6) q^{29} + (3 \beta - 5) q^{31} - 2 q^{35} - 4 q^{37} + 2 \beta q^{41} + \cdots + 2 q^{97} +O(q^{100})$ q + q^5 - 2 * q^7 + (b - 3) * q^11 + q^13 - 2b q^17 + (-3b + 1) q^19 + (b + 3) * q^23 + q^25 + (2b + 6) q^29 + (3b - 5) q^31 - 2 * q^35 - 4 * q^37 + 2b q^41 + (-3b - 5) q^43 + 6 * q^47 - 3 * q^49 + 6b q^53 + (b - 3) * q^55 + (-7b - 3) q^59 + (6b + 2) q^61 + q^65 + (6b + 4) q^67 + (-b + 3) * q^71 - 4 * q^73 + (-2b + 6) q^77 + (-6b - 2) q^79 - 6 * q^83 - 2b q^85 + (-4b + 6) q^89 - 2 * q^91 + (-3b + 1) q^95 + 2 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} - 4 q^{7} - 6 q^{11} + 2 q^{13} + 2 q^{19} + 6 q^{23} + 2 q^{25} + 12 q^{29} - 10 q^{31} - 4 q^{35} - 8 q^{37} - 10 q^{43} + 12 q^{47} - 6 q^{49} - 6 q^{55} - 6 q^{59} + 4 q^{61} + 2 q^{65}+ \cdots + 4 q^{97}+O(q^{100})$ 2 * q + 2 * q^5 - 4 * q^7 - 6 * q^11 + 2 * q^13 + 2 * q^19 + 6 * q^23 + 2 * q^25 + 12 * q^29 - 10 * q^31 - 4 * q^35 - 8 * q^37 - 10 * q^43 + 12 * q^47 - 6 * q^49 - 6 * q^55 - 6 * q^59 + 4 * q^61 + 2 * q^65 + 8 * q^67 + 6 * q^71 - 8 * q^73 + 12 * q^77 - 4 * q^79 - 12 * q^83 + 12 * q^89 - 4 * q^91 + 2 * q^95 + 4 * 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

−1.73205
1.73205

1.00000

−2.00000

1.2

1.00000

−2.00000

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.cm		2
3.b	odd	2	1		1040.2.a.h		2
4.b	odd	2	1		585.2.a.k		2
12.b	even	2	1		65.2.a.c	✓	2
15.d	odd	2	1		5200.2.a.ca		2
20.d	odd	2	1		2925.2.a.z		2
20.e	even	4	2		2925.2.c.v		4
24.f	even	2	1		4160.2.a.y		2
24.h	odd	2	1		4160.2.a.bj		2
52.b	odd	2	1		7605.2.a.be		2
60.h	even	2	1		325.2.a.g		2
60.l	odd	4	2		325.2.b.e		4
84.h	odd	2	1		3185.2.a.k		2
132.d	odd	2	1		7865.2.a.h		2
156.h	even	2	1		845.2.a.d		2
156.l	odd	4	2		845.2.c.e		4
156.p	even	6	2		845.2.e.e		4
156.r	even	6	2		845.2.e.f		4
156.v	odd	12	2		845.2.m.a		4
156.v	odd	12	2		845.2.m.c		4
780.d	even	2	1		4225.2.a.w		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.a.c	✓	2	12.b	even	2	1
325.2.a.g		2	60.h	even	2	1
325.2.b.e		4	60.l	odd	4	2
585.2.a.k		2	4.b	odd	2	1
845.2.a.d		2	156.h	even	2	1
845.2.c.e		4	156.l	odd	4	2
845.2.e.e		4	156.p	even	6	2
845.2.e.f		4	156.r	even	6	2
845.2.m.a		4	156.v	odd	12	2
845.2.m.c		4	156.v	odd	12	2
1040.2.a.h		2	3.b	odd	2	1
2925.2.a.z		2	20.d	odd	2	1
2925.2.c.v		4	20.e	even	4	2
3185.2.a.k		2	84.h	odd	2	1
4160.2.a.y		2	24.f	even	2	1
4160.2.a.bj		2	24.h	odd	2	1
4225.2.a.w		2	780.d	even	2	1
5200.2.a.ca		2	15.d	odd	2	1
7605.2.a.be		2	52.b	odd	2	1
7865.2.a.h		2	132.d	odd	2	1
9360.2.a.cm		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_{11}^{2} + 6T_{11} + 6

T_{17}^{2} - 12

T_{19}^{2} - 2T_{19} - 26

T_{31}^{2} + 10T_{31} - 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$(T + 2)^{2}$ (T + 2)^2
$11$	$T^{2} + 6T + 6$ T^2 + 6*T + 6
$13$	$(T - 1)^{2}$ (T - 1)^2
$17$	$T^{2} - 12$ T^2 - 12
$19$	$T^{2} - 2T - 26$ T^2 - 2*T - 26
$23$	$T^{2} - 6T + 6$ T^2 - 6*T + 6
$29$	$T^{2} - 12T + 24$ T^2 - 12*T + 24
$31$	$T^{2} + 10T - 2$ T^2 + 10*T - 2
$37$	$(T + 4)^{2}$ (T + 4)^2
$41$	$T^{2} - 12$ T^2 - 12
$43$	$T^{2} + 10T - 2$ T^2 + 10*T - 2
$47$	$(T - 6)^{2}$ (T - 6)^2
$53$	$T^{2} - 108$ T^2 - 108
$59$	$T^{2} + 6T - 138$ T^2 + 6*T - 138
$61$	$T^{2} - 4T - 104$ T^2 - 4*T - 104
$67$	$T^{2} - 8T - 92$ T^2 - 8*T - 92
$71$	$T^{2} - 6T + 6$ T^2 - 6*T + 6
$73$	$(T + 4)^{2}$ (T + 4)^2
$79$	$T^{2} + 4T - 104$ T^2 + 4*T - 104
$83$	$(T + 6)^{2}$ (T + 6)^2
$89$	$T^{2} - 12T - 12$ T^2 - 12*T - 12
$97$	$(T - 2)^{2}$ (T - 2)^2
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