LMFDB - Newform orbit 1440.4.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(289,1440)]

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

lf = mfeigenbasis(mf)

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, 4, names="a")

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

chi := DirichletCharacter("1440.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1440 = 2^{5} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1440.f (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:	no
Analytic conductor:	$84.9627504083$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -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 $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - 5 i + 10) q^{5} + 18 i q^{7} - 30 q^{11} + 38 i q^{13} - 70 i q^{17} - 12 q^{19} - 72 i q^{23} + ( - 100 i + 75) q^{25} - 64 q^{29} - 312 q^{31} + (180 i + 90) q^{35} - 138 i q^{37} + 374 q^{41} + \cdots + 376 i q^{97} +O(q^{100})$ q + (-5i + 10) q^5 + 18i q^7 - 30 * q^11 + 38i q^13 - 70i q^17 - 12 * q^19 - 72i q^23 + (-100i + 75) q^25 - 64 * q^29 - 312 * q^31 + (180i + 90) q^35 - 138i q^37 + 374 * q^41 - 468i q^43 + 132i q^47 + 19 * q^49 - 446i q^53 + (150i - 300) q^55 + 510 * q^59 + 754 * q^61 + (380i + 190) q^65 - 384i q^67 + 924 * q^71 - 340i q^73 - 540i q^77 - 72 * q^79 + 156i q^83 + (-700i - 350) q^85 - 290 * q^89 - 684 * q^91 + (60i - 120) q^95 + 376i q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 20 q^{5} - 60 q^{11} - 24 q^{19} + 150 q^{25} - 128 q^{29} - 624 q^{31} + 180 q^{35} + 748 q^{41} + 38 q^{49} - 600 q^{55} + 1020 q^{59} + 1508 q^{61} + 380 q^{65} + 1848 q^{71} - 144 q^{79} - 700 q^{85}+ \cdots - 240 q^{95}+O(q^{100})$ 2 * q + 20 * q^5 - 60 * q^11 - 24 * q^19 + 150 * q^25 - 128 * q^29 - 624 * q^31 + 180 * q^35 + 748 * q^41 + 38 * q^49 - 600 * q^55 + 1020 * q^59 + 1508 * q^61 + 380 * q^65 + 1848 * q^71 - 144 * q^79 - 700 * q^85 - 580 * q^89 - 1368 * q^91 - 240 * q^95

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

10.0000

−

5.00000i

18.0000i

289.2

10.0000

5.00000i

−

18.0000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1440.4.f.c		2
3.b	odd	2	1		480.4.f.b	yes	2
4.b	odd	2	1		1440.4.f.d		2
5.b	even	2	1	inner	1440.4.f.c		2
12.b	even	2	1		480.4.f.a	✓	2
15.d	odd	2	1		480.4.f.b	yes	2
15.e	even	4	1		2400.4.a.j		1
15.e	even	4	1		2400.4.a.n		1
20.d	odd	2	1		1440.4.f.d		2
24.f	even	2	1		960.4.f.k		2
24.h	odd	2	1		960.4.f.h		2
60.h	even	2	1		480.4.f.a	✓	2
60.l	odd	4	1		2400.4.a.i		1
60.l	odd	4	1		2400.4.a.m		1
120.i	odd	2	1		960.4.f.h		2
120.m	even	2	1		960.4.f.k		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
480.4.f.a	✓	2	12.b	even	2	1
480.4.f.a	✓	2	60.h	even	2	1
480.4.f.b	yes	2	3.b	odd	2	1
480.4.f.b	yes	2	15.d	odd	2	1
960.4.f.h		2	24.h	odd	2	1
960.4.f.h		2	120.i	odd	2	1
960.4.f.k		2	24.f	even	2	1
960.4.f.k		2	120.m	even	2	1
1440.4.f.c		2	1.a	even	1	1	trivial
1440.4.f.c		2	5.b	even	2	1	inner
1440.4.f.d		2	4.b	odd	2	1
1440.4.f.d		2	20.d	odd	2	1
2400.4.a.i		1	60.l	odd	4	1
2400.4.a.j		1	15.e	even	4	1
2400.4.a.m		1	60.l	odd	4	1
2400.4.a.n		1	15.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 $S_{4}^{\mathrm{new}}(1440, [\chi])$ :

T_{7}^{2} + 324

T_{11} + 30

T_{17}^{2} + 4900

T_{29} + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} - 20T + 125$ T^2 - 20*T + 125
$7$	$T^{2} + 324$ T^2 + 324
$11$	$(T + 30)^{2}$ (T + 30)^2
$13$	$T^{2} + 1444$ T^2 + 1444
$17$	$T^{2} + 4900$ T^2 + 4900
$19$	$(T + 12)^{2}$ (T + 12)^2
$23$	$T^{2} + 5184$ T^2 + 5184
$29$	$(T + 64)^{2}$ (T + 64)^2
$31$	$(T + 312)^{2}$ (T + 312)^2
$37$	$T^{2} + 19044$ T^2 + 19044
$41$	$(T - 374)^{2}$ (T - 374)^2
$43$	$T^{2} + 219024$ T^2 + 219024
$47$	$T^{2} + 17424$ T^2 + 17424
$53$	$T^{2} + 198916$ T^2 + 198916
$59$	$(T - 510)^{2}$ (T - 510)^2
$61$	$(T - 754)^{2}$ (T - 754)^2
$67$	$T^{2} + 147456$ T^2 + 147456
$71$	$(T - 924)^{2}$ (T - 924)^2
$73$	$T^{2} + 115600$ T^2 + 115600
$79$	$(T + 72)^{2}$ (T + 72)^2
$83$	$T^{2} + 24336$ T^2 + 24336
$89$	$(T + 290)^{2}$ (T + 290)^2
$97$	$T^{2} + 141376$ T^2 + 141376
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