LMFDB - Newform orbit 116.1.d.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [116,1,Mod(115,116)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(116, 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("116.115");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 116 = 2^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	116.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:	$0.0578915414654$
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 3.1.116.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.53824.1
Stark unit:	Root of $x^{6} + x^{5} - 8x^{4} - 17x^{3} - 8x^{2} + x + 1$

$q$-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^{5} - q^{6} - q^{8}+O(q^{10}) $

$ q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{10} + q^{11} + q^{12} - q^{13} - q^{15} + q^{16} - 2 q^{19} - q^{20} - q^{22} - q^{24} + q^{26} - q^{27} + q^{29} + q^{30} + q^{31} - q^{32} + q^{33} + 2 q^{38} - q^{39} + q^{40} + q^{43} + q^{44} + q^{47} + q^{48} + q^{49} - q^{52} - q^{53} + q^{54} - q^{55} - 2 q^{57} - q^{58} - q^{60} - q^{62} + q^{64} + q^{65} - q^{66} - 2 q^{76} + q^{78} + q^{79} - q^{80} - q^{81} - q^{86} + q^{87} - q^{88} + q^{93} - q^{94} + 2 q^{95} - q^{96} - q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/116\mathbb{Z}\right)^\times$.

$n$	$59$	$89$
$\chi(n)$	$1$	$1$

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} $

115.1

−1.00000

1.00000

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
116.d	odd	2	1	CM by $\Q(\sqrt{-29}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	116.1.d.a	✓	1
3.b	odd	2	1		1044.1.g.b		1
4.b	odd	2	1		116.1.d.b	yes	1
5.b	even	2	1		2900.1.g.d		1
5.c	odd	4	2		2900.1.e.b		2
8.b	even	2	1		1856.1.h.a		1
8.d	odd	2	1		1856.1.h.c		1
12.b	even	2	1		1044.1.g.a		1
20.d	odd	2	1		2900.1.g.a		1
20.e	even	4	2		2900.1.e.a		2
29.b	even	2	1		116.1.d.b	yes	1
29.c	odd	4	2		3364.1.b.a		2
29.d	even	7	6		3364.1.h.b		6
29.e	even	14	6		3364.1.h.a		6
29.f	odd	28	12		3364.1.j.f		12
87.d	odd	2	1		1044.1.g.a		1
116.d	odd	2	1	CM	116.1.d.a	✓	1
116.e	even	4	2		3364.1.b.a		2
116.h	odd	14	6		3364.1.h.b		6
116.j	odd	14	6		3364.1.h.a		6
116.l	even	28	12		3364.1.j.f		12
145.d	even	2	1		2900.1.g.a		1
145.h	odd	4	2		2900.1.e.a		2
232.b	odd	2	1		1856.1.h.a		1
232.g	even	2	1		1856.1.h.c		1
348.b	even	2	1		1044.1.g.b		1
580.e	odd	2	1		2900.1.g.d		1
580.o	even	4	2		2900.1.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
116.1.d.a	✓	1	1.a	even	1	1	trivial
116.1.d.a	✓	1	116.d	odd	2	1	CM
116.1.d.b	yes	1	4.b	odd	2	1
116.1.d.b	yes	1	29.b	even	2	1
1044.1.g.a		1	12.b	even	2	1
1044.1.g.a		1	87.d	odd	2	1
1044.1.g.b		1	3.b	odd	2	1
1044.1.g.b		1	348.b	even	2	1
1856.1.h.a		1	8.b	even	2	1
1856.1.h.a		1	232.b	odd	2	1
1856.1.h.c		1	8.d	odd	2	1
1856.1.h.c		1	232.g	even	2	1
2900.1.e.a		2	20.e	even	4	2
2900.1.e.a		2	145.h	odd	4	2
2900.1.e.b		2	5.c	odd	4	2
2900.1.e.b		2	580.o	even	4	2
2900.1.g.a		1	20.d	odd	2	1
2900.1.g.a		1	145.d	even	2	1
2900.1.g.d		1	5.b	even	2	1
2900.1.g.d		1	580.e	odd	2	1
3364.1.b.a		2	29.c	odd	4	2
3364.1.b.a		2	116.e	even	4	2
3364.1.h.a		6	29.e	even	14	6
3364.1.h.a		6	116.j	odd	14	6
3364.1.h.b		6	29.d	even	7	6
3364.1.h.b		6	116.h	odd	14	6
3364.1.j.f		12	29.f	odd	28	12
3364.1.j.f		12	116.l	even	28	12

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 1 $ T3 - 1 acting on $S_{1}^{\mathrm{new}}(116, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T - 1 $ T - 1
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T - 1 $ T - 1
$13$	$ T + 1 $ T + 1
$17$	$ T $ T
$19$	$ T + 2 $ T + 2
$23$	$ T $ T
$29$	$ T - 1 $ T - 1
$31$	$ T - 1 $ T - 1
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T - 1 $ T - 1
$47$	$ T - 1 $ T - 1
$53$	$ T + 1 $ T + 1
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T - 1 $ T - 1
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 116 = 2^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	116.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(59\)	\(89\)
\(\chi(n)\)	\(1\)	\(1\)

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