LMFDB - Newform orbit 3920.1.bt.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3920,1,Mod(79,3920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3920, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 2])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3920.79"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$3920 = 2^{4} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3920.bt (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.95633484952$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(i, \sqrt{35})$
Artin image:	$D_4:C_6$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

$q$ -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.

$f(q)$	$=$	$q - \zeta_{12} q^{5} - \zeta_{12}^{4} q^{9} - 2 \zeta_{12}^{3} q^{13} + 2 \zeta_{12}^{5} q^{17} + \zeta_{12}^{2} q^{25} - 2 q^{29} + \zeta_{12}^{5} q^{45} + 2 \zeta_{12}^{4} q^{65} + 2 \zeta_{12}^{5} q^{73} + \cdots - 2 \zeta_{12}^{3} q^{97} +O(q^{100})$ q - z * q^5 - z^4 * q^9 - 2z^3 q^13 + 2z^5 q^17 + z^2 * q^25 - 2 * q^29 + z^5 * q^45 + 2z^4 q^65 + 2z^5 q^73 - z^2 * q^81 + 2 * q^85 - 2z^3 q^97
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{9} + 2 q^{25} - 8 q^{29} - 4 q^{65} - 2 q^{81} + 8 q^{85}+O(q^{100})$ 4 * q + 2 * q^9 + 2 * q^25 - 8 * q^29 - 4 * q^65 - 2 * q^81 + 8 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$ .

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$\zeta_{12}^{4}$	$-1$

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}

79.1

0.866025	+	0.500000i
−0.866025	−	0.500000i
0.866025	−	0.500000i
−0.866025	+	0.500000i

−0.866025

−

0.500000i

0.500000

−

0.866025i

79.2

0.866025

0.500000i

0.500000

−

0.866025i

1439.1

−0.866025

0.500000i

0.500000

0.866025i

1439.2

0.866025

−

0.500000i

0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
35.c	odd	2	1	CM by $\Q(\sqrt{-35})$
140.c	even	2	1	RM by $\Q(\sqrt{35})$
5.b	even	2	1	inner
7.b	odd	2	1	inner
7.c	even	3	1	inner
7.d	odd	6	1	inner
20.d	odd	2	1	inner
28.d	even	2	1	inner
28.f	even	6	1	inner
28.g	odd	6	1	inner
35.i	odd	6	1	inner
35.j	even	6	1	inner
140.p	odd	6	1	inner
140.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3920.1.bt.f		4
4.b	odd	2	1	CM	3920.1.bt.f		4
5.b	even	2	1	inner	3920.1.bt.f		4
7.b	odd	2	1	inner	3920.1.bt.f		4
7.c	even	3	1		3920.1.j.c	✓	2
7.c	even	3	1	inner	3920.1.bt.f		4
7.d	odd	6	1		3920.1.j.c	✓	2
7.d	odd	6	1	inner	3920.1.bt.f		4
20.d	odd	2	1	inner	3920.1.bt.f		4
28.d	even	2	1	inner	3920.1.bt.f		4
28.f	even	6	1		3920.1.j.c	✓	2
28.f	even	6	1	inner	3920.1.bt.f		4
28.g	odd	6	1		3920.1.j.c	✓	2
28.g	odd	6	1	inner	3920.1.bt.f		4
35.c	odd	2	1	CM	3920.1.bt.f		4
35.i	odd	6	1		3920.1.j.c	✓	2
35.i	odd	6	1	inner	3920.1.bt.f		4
35.j	even	6	1		3920.1.j.c	✓	2
35.j	even	6	1	inner	3920.1.bt.f		4
140.c	even	2	1	RM	3920.1.bt.f		4
140.p	odd	6	1		3920.1.j.c	✓	2
140.p	odd	6	1	inner	3920.1.bt.f		4
140.s	even	6	1		3920.1.j.c	✓	2
140.s	even	6	1	inner	3920.1.bt.f		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3920.1.j.c	✓	2	7.c	even	3	1
3920.1.j.c	✓	2	7.d	odd	6	1
3920.1.j.c	✓	2	28.f	even	6	1
3920.1.j.c	✓	2	28.g	odd	6	1
3920.1.j.c	✓	2	35.i	odd	6	1
3920.1.j.c	✓	2	35.j	even	6	1
3920.1.j.c	✓	2	140.p	odd	6	1
3920.1.j.c	✓	2	140.s	even	6	1
3920.1.bt.f		4	1.a	even	1	1	trivial
3920.1.bt.f		4	4.b	odd	2	1	CM
3920.1.bt.f		4	5.b	even	2	1	inner
3920.1.bt.f		4	7.b	odd	2	1	inner
3920.1.bt.f		4	7.c	even	3	1	inner
3920.1.bt.f		4	7.d	odd	6	1	inner
3920.1.bt.f		4	20.d	odd	2	1	inner
3920.1.bt.f		4	28.d	even	2	1	inner
3920.1.bt.f		4	28.f	even	6	1	inner
3920.1.bt.f		4	28.g	odd	6	1	inner
3920.1.bt.f		4	35.c	odd	2	1	CM
3920.1.bt.f		4	35.i	odd	6	1	inner
3920.1.bt.f		4	35.j	even	6	1	inner
3920.1.bt.f		4	140.c	even	2	1	RM
3920.1.bt.f		4	140.p	odd	6	1	inner
3920.1.bt.f		4	140.s	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3920, [\chi])$ :

T_{3}

T3

T_{11}

T11

T_{13}^{2} + 4

T13^2 + 4

T_{41}

T41

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4}$ T^4
$5$	$T^{4} - T^{2} + 1$ T^4 - T^2 + 1
$7$	$T^{4}$ T^4
$11$	$T^{4}$ T^4
$13$	$(T^{2} + 4)^{2}$ (T^2 + 4)^2
$17$	$T^{4} - 4T^{2} + 16$ T^4 - 4*T^2 + 16
$19$	$T^{4}$ T^4
$23$	$T^{4}$ T^4
$29$	$(T + 2)^{4}$ (T + 2)^4
$31$	$T^{4}$ T^4
$37$	$T^{4}$ T^4
$41$	$T^{4}$ T^4
$43$	$T^{4}$ T^4
$47$	$T^{4}$ T^4
$53$	$T^{4}$ T^4
$59$	$T^{4}$ T^4
$61$	$T^{4}$ T^4
$67$	$T^{4}$ T^4
$71$	$T^{4}$ T^4
$73$	$T^{4} - 4T^{2} + 16$ T^4 - 4*T^2 + 16
$79$	$T^{4}$ T^4
$83$	$T^{4}$ T^4
$89$	$T^{4}$ T^4
$97$	$(T^{2} + 4)^{2}$ (T^2 + 4)^2
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Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3920.1.bt.f

Level	$3920$
Weight	$1$
Character orbit	3920.bt
Analytic conductor	$1.956$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{2}$
CM/RM discs	-4, -35, 140
Inner twists	$16$