LMFDB - Newform orbit 585.2.a.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,2,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("585.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$585 = 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	585.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:	$4.67124851824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 4$ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
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 $\beta = \frac{1}{2}(1 + \sqrt{17})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{2} + (\beta + 2) q^{4} + q^{5} + (\beta - 3) q^{7} + (\beta + 4) q^{8} + \beta q^{10} + ( - \beta + 1) q^{11} - q^{13} + ( - 2 \beta + 4) q^{14} + 3 \beta q^{16} + (\beta - 1) q^{17} - 2 \beta q^{19} + \cdots + (\beta - 20) q^{98} +O(q^{100})$ q + b * q^2 + (b + 2) * q^4 + q^5 + (b - 3) * q^7 + (b + 4) * q^8 + b * q^10 + (-b + 1) * q^11 - q^13 + (-2b + 4) q^14 + 3b q^16 + (b - 1) * q^17 - 2b q^19 + (b + 2) * q^20 - 4 * q^22 + (-b + 5) * q^23 + q^25 - b * q^26 - 2 * q^28 + (-2b - 2) q^29 + 6 * q^31 + (b + 4) * q^32 + 4 * q^34 + (b - 3) * q^35 + (-3b - 3) q^37 + (-2b - 8) q^38 + (b + 4) * q^40 + (b + 1) * q^41 + (2b - 2) q^43 + (-2b - 2) q^44 + (4b - 4) q^46 + (2b + 6) q^47 + (-5b + 6) q^49 + b * q^50 + (-b - 2) * q^52 + (-3b + 3) q^53 + (-b + 1) * q^55 + (2b - 8) q^56 + (-4b - 8) q^58 + 12 * q^59 + (-3b + 1) q^61 + 6b q^62 + (-b + 4) * q^64 - q^65 + (-6b + 4) q^67 + (2b + 2) q^68 + (-2b + 4) q^70 + (-b + 13) * q^71 - 6 * q^73 + (-6b - 12) q^74 + (-6b - 8) q^76 + (3b - 7) q^77 + (3b - 3) q^79 + 3b q^80 + (2b + 4) q^82 + (-8b + 4) q^83 + (b - 1) * q^85 + 8 * q^86 - 4b q^88 + (3b + 3) q^89 + (-b + 3) * q^91 + (2b + 6) q^92 + (8b + 8) q^94 - 2b q^95 + (7b - 1) q^97 + (b - 20) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} + 5 q^{4} + 2 q^{5} - 5 q^{7} + 9 q^{8} + q^{10} + q^{11} - 2 q^{13} + 6 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} + 5 q^{20} - 8 q^{22} + 9 q^{23} + 2 q^{25} - q^{26} - 4 q^{28} - 6 q^{29}+ \cdots - 39 q^{98}+O(q^{100})$ 2 * q + q^2 + 5 * q^4 + 2 * q^5 - 5 * q^7 + 9 * q^8 + q^10 + q^11 - 2 * q^13 + 6 * q^14 + 3 * q^16 - q^17 - 2 * q^19 + 5 * q^20 - 8 * q^22 + 9 * q^23 + 2 * q^25 - q^26 - 4 * q^28 - 6 * q^29 + 12 * q^31 + 9 * q^32 + 8 * q^34 - 5 * q^35 - 9 * q^37 - 18 * q^38 + 9 * q^40 + 3 * q^41 - 2 * q^43 - 6 * q^44 - 4 * q^46 + 14 * q^47 + 7 * q^49 + q^50 - 5 * q^52 + 3 * q^53 + q^55 - 14 * q^56 - 20 * q^58 + 24 * q^59 - q^61 + 6 * q^62 + 7 * q^64 - 2 * q^65 + 2 * q^67 + 6 * q^68 + 6 * q^70 + 25 * q^71 - 12 * q^73 - 30 * q^74 - 22 * q^76 - 11 * q^77 - 3 * q^79 + 3 * q^80 + 10 * q^82 - q^85 + 16 * q^86 - 4 * q^88 + 9 * q^89 + 5 * q^91 + 14 * q^92 + 24 * q^94 - 2 * q^95 + 5 * q^97 - 39 * q^98

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

−1.56155
2.56155

−1.56155

0.438447

1.00000

−4.56155

2.43845

−1.56155

1.2

2.56155

4.56155

1.00000

−0.438447

6.56155

2.56155

Atkin-Lehner signs

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

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	585.2.a.l	yes	2
3.b	odd	2	1		585.2.a.j	✓	2
4.b	odd	2	1		9360.2.a.cw		2
5.b	even	2	1		2925.2.a.x		2
5.c	odd	4	2		2925.2.c.p		4
12.b	even	2	1		9360.2.a.cl		2
13.b	even	2	1		7605.2.a.bd		2
15.d	odd	2	1		2925.2.a.bc		2
15.e	even	4	2		2925.2.c.o		4
39.d	odd	2	1		7605.2.a.bi		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
585.2.a.j	✓	2	3.b	odd	2	1
585.2.a.l	yes	2	1.a	even	1	1	trivial
2925.2.a.x		2	5.b	even	2	1
2925.2.a.bc		2	15.d	odd	2	1
2925.2.c.o		4	15.e	even	4	2
2925.2.c.p		4	5.c	odd	4	2
7605.2.a.bd		2	13.b	even	2	1
7605.2.a.bi		2	39.d	odd	2	1
9360.2.a.cl		2	12.b	even	2	1
9360.2.a.cw		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(585))$ :

T_{2}^{2} - T_{2} - 4

T_{7}^{2} + 5T_{7} + 2

T_{11}^{2} - T_{11} - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - T - 4$ T^2 - T - 4
$3$	$T^{2}$ T^2
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$T^{2} + 5T + 2$ T^2 + 5*T + 2
$11$	$T^{2} - T - 4$ T^2 - T - 4
$13$	$(T + 1)^{2}$ (T + 1)^2
$17$	$T^{2} + T - 4$ T^2 + T - 4
$19$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$23$	$T^{2} - 9T + 16$ T^2 - 9*T + 16
$29$	$T^{2} + 6T - 8$ T^2 + 6*T - 8
$31$	$(T - 6)^{2}$ (T - 6)^2
$37$	$T^{2} + 9T - 18$ T^2 + 9*T - 18
$41$	$T^{2} - 3T - 2$ T^2 - 3*T - 2
$43$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$47$	$T^{2} - 14T + 32$ T^2 - 14*T + 32
$53$	$T^{2} - 3T - 36$ T^2 - 3*T - 36
$59$	$(T - 12)^{2}$ (T - 12)^2
$61$	$T^{2} + T - 38$ T^2 + T - 38
$67$	$T^{2} - 2T - 152$ T^2 - 2*T - 152
$71$	$T^{2} - 25T + 152$ T^2 - 25*T + 152
$73$	$(T + 6)^{2}$ (T + 6)^2
$79$	$T^{2} + 3T - 36$ T^2 + 3*T - 36
$83$	$T^{2} - 272$ T^2 - 272
$89$	$T^{2} - 9T - 18$ T^2 - 9*T - 18
$97$	$T^{2} - 5T - 202$ T^2 - 5*T - 202
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 $\F_{q}$
Belyi maps

$n$ :		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