LMFDB - Newform orbit 3332.1.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(883,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("3332.883");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3332 = 2^{2} \cdot 7^{2} \cdot 17$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3332.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:	$1.66288462209$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 68)
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(i, \sqrt{17})$
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.13328.1
Stark unit:	Root of $x^{4} - 725x^{3} + 1252x^{2} - 725x + 1$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$ .

$n$	$785$	$885$	$1667$
$\chi(n)$	$-1$	$1$	$-1$

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}

883.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
17.b	even	2	1	RM by $\Q(\sqrt{17})$
68.d	odd	2	1	CM by $\Q(\sqrt{-17})$

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3332.1.g.a		1
4.b	odd	2	1	CM	3332.1.g.a		1
7.b	odd	2	1		68.1.d.a	✓	1
7.c	even	3	2		3332.1.o.d		2
7.d	odd	6	2		3332.1.o.c		2
17.b	even	2	1	RM	3332.1.g.a		1
21.c	even	2	1		612.1.e.a		1
28.d	even	2	1		68.1.d.a	✓	1
28.f	even	6	2		3332.1.o.c		2
28.g	odd	6	2		3332.1.o.d		2
35.c	odd	2	1		1700.1.h.d		1
35.f	even	4	2		1700.1.d.b		2
56.e	even	2	1		1088.1.g.a		1
56.h	odd	2	1		1088.1.g.a		1
68.d	odd	2	1	CM	3332.1.g.a		1
84.h	odd	2	1		612.1.e.a		1
119.d	odd	2	1		68.1.d.a	✓	1
119.f	odd	4	2		1156.1.c.a		1
119.h	odd	6	2		3332.1.o.c		2
119.j	even	6	2		3332.1.o.d		2
119.l	odd	8	4		1156.1.f.a		2
119.p	even	16	8		1156.1.g.a		4
140.c	even	2	1		1700.1.h.d		1
140.j	odd	4	2		1700.1.d.b		2
357.c	even	2	1		612.1.e.a		1
476.e	even	2	1		68.1.d.a	✓	1
476.k	even	4	2		1156.1.c.a		1
476.o	odd	6	2		3332.1.o.d		2
476.q	even	6	2		3332.1.o.c		2
476.w	even	8	4		1156.1.f.a		2
476.bf	odd	16	8		1156.1.g.a		4
595.b	odd	2	1		1700.1.h.d		1
595.p	even	4	2		1700.1.d.b		2
952.e	odd	2	1		1088.1.g.a		1
952.k	even	2	1		1088.1.g.a		1
1428.b	odd	2	1		612.1.e.a		1
2380.p	even	2	1		1700.1.h.d		1
2380.bi	odd	4	2		1700.1.d.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
68.1.d.a	✓	1	7.b	odd	2	1
68.1.d.a	✓	1	28.d	even	2	1
68.1.d.a	✓	1	119.d	odd	2	1
68.1.d.a	✓	1	476.e	even	2	1
612.1.e.a		1	21.c	even	2	1
612.1.e.a		1	84.h	odd	2	1
612.1.e.a		1	357.c	even	2	1
612.1.e.a		1	1428.b	odd	2	1
1088.1.g.a		1	56.e	even	2	1
1088.1.g.a		1	56.h	odd	2	1
1088.1.g.a		1	952.e	odd	2	1
1088.1.g.a		1	952.k	even	2	1
1156.1.c.a		1	119.f	odd	4	2
1156.1.c.a		1	476.k	even	4	2
1156.1.f.a		2	119.l	odd	8	4
1156.1.f.a		2	476.w	even	8	4
1156.1.g.a		4	119.p	even	16	8
1156.1.g.a		4	476.bf	odd	16	8
1700.1.d.b		2	35.f	even	4	2
1700.1.d.b		2	140.j	odd	4	2
1700.1.d.b		2	595.p	even	4	2
1700.1.d.b		2	2380.bi	odd	4	2
1700.1.h.d		1	35.c	odd	2	1
1700.1.h.d		1	140.c	even	2	1
1700.1.h.d		1	595.b	odd	2	1
1700.1.h.d		1	2380.p	even	2	1
3332.1.g.a		1	1.a	even	1	1	trivial
3332.1.g.a		1	4.b	odd	2	1	CM
3332.1.g.a		1	17.b	even	2	1	RM
3332.1.g.a		1	68.d	odd	2	1	CM
3332.1.o.c		2	7.d	odd	6	2
3332.1.o.c		2	28.f	even	6	2
3332.1.o.c		2	119.h	odd	6	2
3332.1.o.c		2	476.q	even	6	2
3332.1.o.d		2	7.c	even	3	2
3332.1.o.d		2	28.g	odd	6	2
3332.1.o.d		2	119.j	even	6	2
3332.1.o.d		2	476.o	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3332, [\chi])$ :

T_{3}

T_{5}

T_{11}

T_{13} - 2

Hecke characteristic polynomials

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