LMFDB - Newform orbit 1280.4.d.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1280 = 2^{8} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1280.d (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:	$75.5224448073$
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:	$1$
Twist minimal:	no (minimal twist has level 640)
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.

$f(q)$	$=$	$q + 8 i q^{3} - 5 i q^{5} - 2 q^{7} - 37 q^{9} - 22 i q^{11} + 10 i q^{13} + 40 q^{15} + 10 q^{17} + 110 i q^{19} - 16 i q^{21} + 154 q^{23} - 25 q^{25} - 80 i q^{27} + 222 i q^{29} - 92 q^{31} + 176 q^{33} + \cdots + 814 i q^{99} +O(q^{100})$ q + 8i q^3 - 5i q^5 - 2 * q^7 - 37 * q^9 - 22i q^11 + 10i q^13 + 40 * q^15 + 10 * q^17 + 110i q^19 - 16i q^21 + 154 * q^23 - 25 * q^25 - 80i q^27 + 222i q^29 - 92 * q^31 + 176 * q^33 + 10i q^35 + 34i q^37 - 80 * q^39 - 398 * q^41 - 268i q^43 + 185i q^45 - 10 * q^47 - 339 * q^49 + 80i q^51 + 582i q^53 - 110 * q^55 - 880 * q^57 - 746i q^59 - 226i q^61 + 74 * q^63 + 50 * q^65 - 172i q^67 + 1232i q^69 - 928 * q^71 - 570 * q^73 - 200i q^75 + 44i q^77 + 64 * q^79 - 359 * q^81 - 864i q^83 - 50i q^85 - 1776 * q^87 + 874 * q^89 - 20i q^91 - 736i q^93 + 550 * q^95 + 306 * q^97 + 814i q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{7} - 74 q^{9} + 80 q^{15} + 20 q^{17} + 308 q^{23} - 50 q^{25} - 184 q^{31} + 352 q^{33} - 160 q^{39} - 796 q^{41} - 20 q^{47} - 678 q^{49} - 220 q^{55} - 1760 q^{57} + 148 q^{63} + 100 q^{65}+ \cdots + 612 q^{97}+O(q^{100})$ 2 * q - 4 * q^7 - 74 * q^9 + 80 * q^15 + 20 * q^17 + 308 * q^23 - 50 * q^25 - 184 * q^31 + 352 * q^33 - 160 * q^39 - 796 * q^41 - 20 * q^47 - 678 * q^49 - 220 * q^55 - 1760 * q^57 + 148 * q^63 + 100 * q^65 - 1856 * q^71 - 1140 * q^73 + 128 * q^79 - 718 * q^81 - 3552 * q^87 + 1748 * q^89 + 1100 * q^95 + 612 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$ .

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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}

641.1

	−	1.00000i
		1.00000i

−

8.00000i

5.00000i

−2.00000

−37.0000

641.2

8.00000i

−

5.00000i

−2.00000

−37.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1280.4.d.h		2
4.b	odd	2	1		1280.4.d.i		2
8.b	even	2	1	inner	1280.4.d.h		2
8.d	odd	2	1		1280.4.d.i		2
16.e	even	4	1		640.4.a.a	✓	1
16.e	even	4	1		640.4.a.d	yes	1
16.f	odd	4	1		640.4.a.b	yes	1
16.f	odd	4	1		640.4.a.c	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
640.4.a.a	✓	1	16.e	even	4	1
640.4.a.b	yes	1	16.f	odd	4	1
640.4.a.c	yes	1	16.f	odd	4	1
640.4.a.d	yes	1	16.e	even	4	1
1280.4.d.h		2	1.a	even	1	1	trivial
1280.4.d.h		2	8.b	even	2	1	inner
1280.4.d.i		2	4.b	odd	2	1
1280.4.d.i		2	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(1280, [\chi])$ :

T_{3}^{2} + 64

T_{7} + 2

T_{11}^{2} + 484

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 64$ T^2 + 64
$5$	$T^{2} + 25$ T^2 + 25
$7$	$(T + 2)^{2}$ (T + 2)^2
$11$	$T^{2} + 484$ T^2 + 484
$13$	$T^{2} + 100$ T^2 + 100
$17$	$(T - 10)^{2}$ (T - 10)^2
$19$	$T^{2} + 12100$ T^2 + 12100
$23$	$(T - 154)^{2}$ (T - 154)^2
$29$	$T^{2} + 49284$ T^2 + 49284
$31$	$(T + 92)^{2}$ (T + 92)^2
$37$	$T^{2} + 1156$ T^2 + 1156
$41$	$(T + 398)^{2}$ (T + 398)^2
$43$	$T^{2} + 71824$ T^2 + 71824
$47$	$(T + 10)^{2}$ (T + 10)^2
$53$	$T^{2} + 338724$ T^2 + 338724
$59$	$T^{2} + 556516$ T^2 + 556516
$61$	$T^{2} + 51076$ T^2 + 51076
$67$	$T^{2} + 29584$ T^2 + 29584
$71$	$(T + 928)^{2}$ (T + 928)^2
$73$	$(T + 570)^{2}$ (T + 570)^2
$79$	$(T - 64)^{2}$ (T - 64)^2
$83$	$T^{2} + 746496$ T^2 + 746496
$89$	$(T - 874)^{2}$ (T - 874)^2
$97$	$(T - 306)^{2}$ (T - 306)^2
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