LMFDB - Newform orbit 135.1.d.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,1,Mod(134,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("135.134");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	135.d (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:	$0.0673737767055$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.135.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.54675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + q^{2} - q^{5} - q^{8}+O(q^{10}) $

$ q + q^{2} - q^{5} - q^{8} - q^{10} - q^{16} + q^{17} - q^{19} + q^{23} + q^{25} - q^{31} + q^{34} - q^{38} + q^{40} + q^{46} - 2 q^{47} + q^{49} + q^{50} + q^{53} - q^{61} - q^{62} + q^{64} - q^{79} + q^{80} + q^{83} - q^{85} - 2 q^{94} + q^{95} + q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$-1$	$-1$

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} $

134.1

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	135.1.d.b	yes	1
3.b	odd	2	1		135.1.d.a	✓	1
4.b	odd	2	1		2160.1.c.a		1
5.b	even	2	1		135.1.d.a	✓	1
5.c	odd	4	2		675.1.c.c		2
9.c	even	3	2		405.1.h.a		2
9.d	odd	6	2		405.1.h.b		2
12.b	even	2	1		2160.1.c.b		1
15.d	odd	2	1	CM	135.1.d.b	yes	1
15.e	even	4	2		675.1.c.c		2
20.d	odd	2	1		2160.1.c.b		1
27.e	even	9	6		3645.1.n.e		6
27.f	odd	18	6		3645.1.n.d		6
45.h	odd	6	2		405.1.h.a		2
45.j	even	6	2		405.1.h.b		2
45.k	odd	12	4		2025.1.j.c		4
45.l	even	12	4		2025.1.j.c		4
60.h	even	2	1		2160.1.c.a		1
135.n	odd	18	6		3645.1.n.e		6
135.p	even	18	6		3645.1.n.d		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
135.1.d.a	✓	1	3.b	odd	2	1
135.1.d.a	✓	1	5.b	even	2	1
135.1.d.b	yes	1	1.a	even	1	1	trivial
135.1.d.b	yes	1	15.d	odd	2	1	CM
405.1.h.a		2	9.c	even	3	2
405.1.h.a		2	45.h	odd	6	2
405.1.h.b		2	9.d	odd	6	2
405.1.h.b		2	45.j	even	6	2
675.1.c.c		2	5.c	odd	4	2
675.1.c.c		2	15.e	even	4	2
2025.1.j.c		4	45.k	odd	12	4
2025.1.j.c		4	45.l	even	12	4
2160.1.c.a		1	4.b	odd	2	1
2160.1.c.a		1	60.h	even	2	1
2160.1.c.b		1	12.b	even	2	1
2160.1.c.b		1	20.d	odd	2	1
3645.1.n.d		6	27.f	odd	18	6
3645.1.n.d		6	135.p	even	18	6
3645.1.n.e		6	27.e	even	9	6
3645.1.n.e		6	135.n	odd	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} - 1 $ T2 - 1 acting on $S_{1}^{\mathrm{new}}(135, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T - 1 $ T - 1
$19$	$ T + 1 $ T + 1
$23$	$ T - 1 $ T - 1
$29$	$ T $ T
$31$	$ T + 1 $ T + 1
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T + 2 $ T + 2
$53$	$ T - 1 $ T - 1
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T + 1 $ T + 1
$83$	$ T - 1 $ T - 1
$89$	$ T $ T
$97$	$ T $ T
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	135.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(56\)	\(82\)
\(\chi(n)\)	\(-1\)	\(-1\)

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