LMFDB - Newform orbit 1988.1.g.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1988,1,Mod(1987,1988)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1988, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1988.1987");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1988 = 2^{2} \cdot 7 \cdot 71 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1988.g (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:	$0.992141245114$
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 3.1.1988.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.15808576.1

$q$-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^{3} + q^{4} + q^{6} - q^{7} + q^{8}+O(q^{10}) $

$ q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{11} + q^{12} - q^{13} - q^{14} + q^{16} + 2 q^{17} - 2 q^{19} - q^{21} + q^{22} + q^{23} + q^{24} + q^{25} - q^{26} - q^{27} - q^{28} - q^{29} + q^{32} + q^{33} + 2 q^{34} - q^{37} - 2 q^{38} - q^{39} - q^{41} - q^{42} + q^{44} + q^{46} + q^{48} + q^{49} + q^{50} + 2 q^{51} - q^{52} - q^{54} - q^{56} - 2 q^{57} - q^{58} - q^{61} + q^{64} + q^{66} - 2 q^{67} + 2 q^{68} + q^{69} - q^{71} - q^{74} + q^{75} - 2 q^{76} - q^{77} - q^{78} - q^{81} - q^{82} + q^{83} - q^{84} - q^{87} + q^{88} + q^{91} + q^{92} + q^{96} - q^{97} + q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1988\mathbb{Z}\right)^\times$.

$n$	$995$	$1137$	$1569$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

1987.1

1.00000

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
1988.g	odd	2	1	CM by $\Q(\sqrt{-497}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1988.1.g.e	yes	1
4.b	odd	2	1		1988.1.g.d	yes	1
7.b	odd	2	1		1988.1.g.c	✓	1
28.d	even	2	1		1988.1.g.f	yes	1
71.b	odd	2	1		1988.1.g.f	yes	1
284.c	even	2	1		1988.1.g.c	✓	1
497.b	even	2	1		1988.1.g.d	yes	1
1988.g	odd	2	1	CM	1988.1.g.e	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1988.1.g.c	✓	1	7.b	odd	2	1
1988.1.g.c	✓	1	284.c	even	2	1
1988.1.g.d	yes	1	4.b	odd	2	1
1988.1.g.d	yes	1	497.b	even	2	1
1988.1.g.e	yes	1	1.a	even	1	1	trivial
1988.1.g.e	yes	1	1988.g	odd	2	1	CM
1988.1.g.f	yes	1	28.d	even	2	1
1988.1.g.f	yes	1	71.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 $S_{1}^{\mathrm{new}}(1988, [\chi])$:

$ T_{3} - 1 $

$ T_{11} - 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T - 1 $ T - 1
$5$	$ T $ T
$7$	$ T + 1 $ T + 1
$11$	$ T - 1 $ T - 1
$13$	$ T + 1 $ T + 1
$17$	$ T - 2 $ T - 2
$19$	$ T + 2 $ T + 2
$23$	$ T - 1 $ T - 1
$29$	$ T + 1 $ T + 1
$31$	$ T $ T
$37$	$ T + 1 $ T + 1
$41$	$ T + 1 $ T + 1
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T + 2 $ T + 2
$71$	$ T + 1 $ T + 1
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T - 1 $ T - 1
$89$	$ T $ T
$97$	$ T + 1 $ T + 1
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Level:	\( N \)	\(=\)	\( 1988 = 2^{2} \cdot 7 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1988.g (of order \(2\), degree \(1\), minimal)

\(n\)	\(995\)	\(1137\)	\(1569\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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