LMFDB - Newform orbit 1440.2.f.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,2,Mod(289,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.289"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1440 = 2^{5} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1440.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
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:	$2$
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $\beta = 2i$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta - 1) q^{5} + 2 \beta q^{13} + 4 \beta q^{17} + (2 \beta - 3) q^{25} + 10 q^{29} - 6 \beta q^{37} + 10 q^{41} + 7 q^{49} + 2 \beta q^{53} + 10 q^{61} + ( - 2 \beta + 8) q^{65} + 8 \beta q^{73} + \cdots + 4 \beta q^{97} +O(q^{100})$ q + (-b - 1) * q^5 + 2b q^13 + 4b q^17 + (2b - 3) q^25 + 10 * q^29 - 6b q^37 + 10 * q^41 + 7 * q^49 + 2b q^53 + 10 * q^61 + (-2b + 8) q^65 + 8b q^73 + (-4b + 16) q^85 - 10 * q^89 + 4b q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} - 6 q^{25} + 20 q^{29} + 20 q^{41} + 14 q^{49} + 20 q^{61} + 16 q^{65} + 32 q^{85} - 20 q^{89}+O(q^{100})$ 2 * q - 2 * q^5 - 6 * q^25 + 20 * q^29 + 20 * q^41 + 14 * q^49 + 20 * q^61 + 16 * q^65 + 32 * q^85 - 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$ .

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$-1$	$1$	$1$	$1$

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}

289.1

		1.00000i
	−	1.00000i

−1.00000

−

2.00000i

289.2

−1.00000

2.00000i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1440.2.f.c		2
3.b	odd	2	1		160.2.c.a	✓	2
4.b	odd	2	1	CM	1440.2.f.c		2
5.b	even	2	1	inner	1440.2.f.c		2
5.c	odd	4	1		7200.2.a.y		1
5.c	odd	4	1		7200.2.a.bb		1
8.b	even	2	1		2880.2.f.n		2
8.d	odd	2	1		2880.2.f.n		2
12.b	even	2	1		160.2.c.a	✓	2
15.d	odd	2	1		160.2.c.a	✓	2
15.e	even	4	1		800.2.a.e		1
15.e	even	4	1		800.2.a.f		1
20.d	odd	2	1	inner	1440.2.f.c		2
20.e	even	4	1		7200.2.a.y		1
20.e	even	4	1		7200.2.a.bb		1
24.f	even	2	1		320.2.c.a		2
24.h	odd	2	1		320.2.c.a		2
40.e	odd	2	1		2880.2.f.n		2
40.f	even	2	1		2880.2.f.n		2
48.i	odd	4	1		1280.2.f.c		2
48.i	odd	4	1		1280.2.f.d		2
48.k	even	4	1		1280.2.f.c		2
48.k	even	4	1		1280.2.f.d		2
60.h	even	2	1		160.2.c.a	✓	2
60.l	odd	4	1		800.2.a.e		1
60.l	odd	4	1		800.2.a.f		1
120.i	odd	2	1		320.2.c.a		2
120.m	even	2	1		320.2.c.a		2
120.q	odd	4	1		1600.2.a.l		1
120.q	odd	4	1		1600.2.a.m		1
120.w	even	4	1		1600.2.a.l		1
120.w	even	4	1		1600.2.a.m		1
240.t	even	4	1		1280.2.f.c		2
240.t	even	4	1		1280.2.f.d		2
240.bm	odd	4	1		1280.2.f.c		2
240.bm	odd	4	1		1280.2.f.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
160.2.c.a	✓	2	3.b	odd	2	1
160.2.c.a	✓	2	12.b	even	2	1
160.2.c.a	✓	2	15.d	odd	2	1
160.2.c.a	✓	2	60.h	even	2	1
320.2.c.a		2	24.f	even	2	1
320.2.c.a		2	24.h	odd	2	1
320.2.c.a		2	120.i	odd	2	1
320.2.c.a		2	120.m	even	2	1
800.2.a.e		1	15.e	even	4	1
800.2.a.e		1	60.l	odd	4	1
800.2.a.f		1	15.e	even	4	1
800.2.a.f		1	60.l	odd	4	1
1280.2.f.c		2	48.i	odd	4	1
1280.2.f.c		2	48.k	even	4	1
1280.2.f.c		2	240.t	even	4	1
1280.2.f.c		2	240.bm	odd	4	1
1280.2.f.d		2	48.i	odd	4	1
1280.2.f.d		2	48.k	even	4	1
1280.2.f.d		2	240.t	even	4	1
1280.2.f.d		2	240.bm	odd	4	1
1440.2.f.c		2	1.a	even	1	1	trivial
1440.2.f.c		2	4.b	odd	2	1	CM
1440.2.f.c		2	5.b	even	2	1	inner
1440.2.f.c		2	20.d	odd	2	1	inner
1600.2.a.l		1	120.q	odd	4	1
1600.2.a.l		1	120.w	even	4	1
1600.2.a.m		1	120.q	odd	4	1
1600.2.a.m		1	120.w	even	4	1
2880.2.f.n		2	8.b	even	2	1
2880.2.f.n		2	8.d	odd	2	1
2880.2.f.n		2	40.e	odd	2	1
2880.2.f.n		2	40.f	even	2	1
7200.2.a.y		1	5.c	odd	4	1
7200.2.a.y		1	20.e	even	4	1
7200.2.a.bb		1	5.c	odd	4	1
7200.2.a.bb		1	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1440, [\chi])$ :

T_{7}

T7

T_{11}

T11

T_{17}^{2} + 64

T17^2 + 64

T_{19}

T19

T_{29} - 10

T29 - 10

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 2T + 5$ T^2 + 2*T + 5
$7$	$T^{2}$ T^2
$11$	$T^{2}$ T^2
$13$	$T^{2} + 16$ T^2 + 16
$17$	$T^{2} + 64$ T^2 + 64
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$(T - 10)^{2}$ (T - 10)^2
$31$	$T^{2}$ T^2
$37$	$T^{2} + 144$ T^2 + 144
$41$	$(T - 10)^{2}$ (T - 10)^2
$43$	$T^{2}$ T^2
$47$	$T^{2}$ T^2
$53$	$T^{2} + 16$ T^2 + 16
$59$	$T^{2}$ T^2
$61$	$(T - 10)^{2}$ (T - 10)^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$T^{2} + 256$ T^2 + 256
$79$	$T^{2}$ T^2
$83$	$T^{2}$ T^2
$89$	$(T + 10)^{2}$ (T + 10)^2
$97$	$T^{2} + 64$ T^2 + 64
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