LMFDB - Newform orbit 59.19.b.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [59,19,Mod(58,59)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(59, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 19, names="a")

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

chi := DirichletCharacter("59.58");

S:= CuspForms(chi, 19);

N := Newforms(S);

Level:	$N$	$=$	$59$
Weight:	$k$	$=$	$19$
Character orbit:	$[\chi]$	$=$	59.b (of order $2$ , degree $1$ , 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:	yes
Analytic conductor:	$121.177821249$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q + 10810 q^{3} + 262144 q^{4} - 1985254 q^{5} - 50982910 q^{7} - 270564389 q^{9} + 2833776640 q^{12} - 21460595740 q^{15} + 68719476736 q^{16} - 216651752350 q^{17} - 62437037542 q^{19} - 520422424576 q^{20}+ \cdots + 12\!\cdots\!68 q^{95}+O(q^{100})

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/59\mathbb{Z}\right)^\times$ .

$n$	$2$
$\chi(n)$	$-1$

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}

58.1

10810.0

262144.

−1.98525e6

−5.09829e7

−2.70564e8

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.b	odd	2	1	CM by $\Q(\sqrt{-59})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	59.19.b.a	✓	1
59.b	odd	2	1	CM	59.19.b.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.19.b.a	✓	1	1.a	even	1	1	trivial
59.19.b.a	✓	1	59.b	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{19}^{\mathrm{new}}(59, [\chi])$ :

T_{2}

T_{3} - 10810

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 10810$ T - 10810
$5$	$T + 1985254$ T + 1985254
$7$	$T + 50982910$ T + 50982910
$11$	$T$ T
$13$	$T$ T
$17$	$T + 216651752350$ T + 216651752350
$19$	$T + 62437037542$ T + 62437037542
$23$	$T$ T
$29$	$T - 28956785336138$ T - 28956785336138
$31$	$T$ T
$37$	$T$ T
$41$	$T - 652243002714578$ T - 652243002714578
$43$	$T$ T
$47$	$T$ T
$53$	$T - 5739806619558650$ T - 5739806619558650
$59$	$T + 8662995818654939$ T + 8662995818654939
$61$	$T$ T
$67$	$T$ T
$71$	$T + 56\!\cdots\!62$ T + 56563270329694462
$73$	$T$ T
$79$	$T - 18\!\cdots\!62$ T - 186548867397995762
$83$	$T$ T
$89$	$T$ T
$97$	$T$ T
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