LMFDB - Newform orbit 3840.1.c.e

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,1,Mod(3329,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3840.3329");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3840 = 2^{8} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3840.c (of order $2$ , degree $1$ , not 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:	no
Analytic conductor:	$1.91640964851$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1920)
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-5}, \sqrt{-6})$
Artin image:	$D_4:C_2$
Artin field:	Galois closure of 8.0.33973862400.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $q$ -expansion and trace form are shown below.

$f(q)$	$=$	$q - q^{3} - i q^{5} + 2 i q^{7} + q^{9} + i q^{15} - 2 i q^{21} - q^{25} - q^{27} + 2 i q^{29} + 2 q^{35} - i q^{45} - 3 q^{49} + 2 i q^{63} + q^{75} + q^{81} - 2 q^{83} - 2 i q^{87} +O(q^{100})$ q - q^3 - z * q^5 + 2z q^7 + q^9 + z * q^15 - 2z q^21 - q^25 - q^27 + 2z q^29 + 2 * q^35 - z * q^45 - 3 * q^49 + 2z q^63 + q^75 + q^81 - 2 * q^83 - 2z q^87
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} + 2 q^{9} - 2 q^{25} - 2 q^{27} + 4 q^{35} - 6 q^{49} + 2 q^{75} + 2 q^{81} - 4 q^{83}+O(q^{100})$ 2 * q - 2 * q^3 + 2 * q^9 - 2 * q^25 - 2 * q^27 + 4 * q^35 - 6 * q^49 + 2 * q^75 + 2 * q^81 - 4 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$ .

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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}

3329.1

		1.00000i
	−	1.00000i

−1.00000

−

1.00000i

2.00000i

1.00000

3329.2

−1.00000

1.00000i

−

2.00000i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
20.d	odd	2	1	CM by $\Q(\sqrt{-5})$
24.h	odd	2	1	CM by $\Q(\sqrt{-6})$
120.m	even	2	1	RM by $\Q(\sqrt{30})$
8.d	odd	2	1	inner
12.b	even	2	1	inner
15.d	odd	2	1	inner
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3840.1.c.e		2
3.b	odd	2	1		3840.1.c.h		2
4.b	odd	2	1		3840.1.c.h		2
5.b	even	2	1		3840.1.c.h		2
8.b	even	2	1		3840.1.c.h		2
8.d	odd	2	1	inner	3840.1.c.e		2
12.b	even	2	1	inner	3840.1.c.e		2
15.d	odd	2	1	inner	3840.1.c.e		2
16.e	even	4	1		1920.1.i.e	✓	2
16.e	even	4	1		1920.1.i.f	yes	2
16.f	odd	4	1		1920.1.i.e	✓	2
16.f	odd	4	1		1920.1.i.f	yes	2
20.d	odd	2	1	CM	3840.1.c.e		2
24.f	even	2	1		3840.1.c.h		2
24.h	odd	2	1	CM	3840.1.c.e		2
40.e	odd	2	1		3840.1.c.h		2
40.f	even	2	1	inner	3840.1.c.e		2
48.i	odd	4	1		1920.1.i.e	✓	2
48.i	odd	4	1		1920.1.i.f	yes	2
48.k	even	4	1		1920.1.i.e	✓	2
48.k	even	4	1		1920.1.i.f	yes	2
60.h	even	2	1		3840.1.c.h		2
80.k	odd	4	1		1920.1.i.e	✓	2
80.k	odd	4	1		1920.1.i.f	yes	2
80.q	even	4	1		1920.1.i.e	✓	2
80.q	even	4	1		1920.1.i.f	yes	2
120.i	odd	2	1		3840.1.c.h		2
120.m	even	2	1	RM	3840.1.c.e		2
240.t	even	4	1		1920.1.i.e	✓	2
240.t	even	4	1		1920.1.i.f	yes	2
240.bm	odd	4	1		1920.1.i.e	✓	2
240.bm	odd	4	1		1920.1.i.f	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1920.1.i.e	✓	2	16.e	even	4	1
1920.1.i.e	✓	2	16.f	odd	4	1
1920.1.i.e	✓	2	48.i	odd	4	1
1920.1.i.e	✓	2	48.k	even	4	1
1920.1.i.e	✓	2	80.k	odd	4	1
1920.1.i.e	✓	2	80.q	even	4	1
1920.1.i.e	✓	2	240.t	even	4	1
1920.1.i.e	✓	2	240.bm	odd	4	1
1920.1.i.f	yes	2	16.e	even	4	1
1920.1.i.f	yes	2	16.f	odd	4	1
1920.1.i.f	yes	2	48.i	odd	4	1
1920.1.i.f	yes	2	48.k	even	4	1
1920.1.i.f	yes	2	80.k	odd	4	1
1920.1.i.f	yes	2	80.q	even	4	1
1920.1.i.f	yes	2	240.t	even	4	1
1920.1.i.f	yes	2	240.bm	odd	4	1
3840.1.c.e		2	1.a	even	1	1	trivial
3840.1.c.e		2	8.d	odd	2	1	inner
3840.1.c.e		2	12.b	even	2	1	inner
3840.1.c.e		2	15.d	odd	2	1	inner
3840.1.c.e		2	20.d	odd	2	1	CM
3840.1.c.e		2	24.h	odd	2	1	CM
3840.1.c.e		2	40.f	even	2	1	inner
3840.1.c.e		2	120.m	even	2	1	RM
3840.1.c.h		2	3.b	odd	2	1
3840.1.c.h		2	4.b	odd	2	1
3840.1.c.h		2	5.b	even	2	1
3840.1.c.h		2	8.b	even	2	1
3840.1.c.h		2	24.f	even	2	1
3840.1.c.h		2	40.e	odd	2	1
3840.1.c.h		2	60.h	even	2	1
3840.1.c.h		2	120.i	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3840, [\chi])$ :

T_{7}^{2} + 4

T_{23}

T_{31}

T_{53}

T_{83} + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T + 1)^{2}$ (T + 1)^2
$5$	$T^{2} + 1$ T^2 + 1
$7$	$T^{2} + 4$ T^2 + 4
$11$	$T^{2}$ T^2
$13$	$T^{2}$ T^2
$17$	$T^{2}$ T^2
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2} + 4$ T^2 + 4
$31$	$T^{2}$ T^2
$37$	$T^{2}$ T^2
$41$	$T^{2}$ T^2
$43$	$T^{2}$ T^2
$47$	$T^{2}$ T^2
$53$	$T^{2}$ T^2
$59$	$T^{2}$ T^2
$61$	$T^{2}$ T^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$T^{2}$ T^2
$79$	$T^{2}$ T^2
$83$	$(T + 2)^{2}$ (T + 2)^2
$89$	$T^{2}$ T^2
$97$	$T^{2}$ T^2
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