LMFDB - Newform orbit 2800.2.g.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2800,2,Mod(449,2800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2800.449"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$2800 = 2^{4} \cdot 5^{2} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2800.g (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-12,0,10,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$22.3581125660$
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 140)
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.

$f(q)$	$=$	$q + 3 i q^{3} + i q^{7} - 6 q^{9} + 5 q^{11} + 3 i q^{13} - i q^{17} + 6 q^{19} - 3 q^{21} + 6 i q^{23} - 9 i q^{27} + 9 q^{29} + 4 q^{31} + 15 i q^{33} + 2 i q^{37} - 9 q^{39} - 4 q^{41} + 10 i q^{43} + \cdots - 30 q^{99} +O(q^{100})$ q + 3i q^3 + i * q^7 - 6 * q^9 + 5 * q^11 + 3i q^13 - i * q^17 + 6 * q^19 - 3 * q^21 + 6i q^23 - 9i q^27 + 9 * q^29 + 4 * q^31 + 15i q^33 + 2i q^37 - 9 * q^39 - 4 * q^41 + 10i q^43 + i * q^47 - q^49 + 3 * q^51 - 4i q^53 + 18i q^57 - 8 * q^59 - 8 * q^61 - 6i q^63 - 12i q^67 - 18 * q^69 - 8 * q^71 - 2i q^73 + 5i q^77 + 13 * q^79 + 9 * q^81 - 4i q^83 + 27i q^87 - 4 * q^89 - 3 * q^91 + 12i q^93 - 13i q^97 - 30 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 12 q^{9} + 10 q^{11} + 12 q^{19} - 6 q^{21} + 18 q^{29} + 8 q^{31} - 18 q^{39} - 8 q^{41} - 2 q^{49} + 6 q^{51} - 16 q^{59} - 16 q^{61} - 36 q^{69} - 16 q^{71} + 26 q^{79} + 18 q^{81} - 8 q^{89} - 6 q^{91}+ \cdots - 60 q^{99}+O(q^{100})$ 2 * q - 12 * q^9 + 10 * q^11 + 12 * q^19 - 6 * q^21 + 18 * q^29 + 8 * q^31 - 18 * q^39 - 8 * q^41 - 2 * q^49 + 6 * q^51 - 16 * q^59 - 16 * q^61 - 36 * q^69 - 16 * q^71 + 26 * q^79 + 18 * q^81 - 8 * q^89 - 6 * q^91 - 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2800\mathbb{Z}\right)^\times$ .

$n$	$351$	$801$	$2101$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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}

449.1

	−	1.00000i
		1.00000i

−

3.00000i

−

1.00000i

−6.00000

449.2

3.00000i

1.00000i

−6.00000

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	2800.2.g.c		2
4.b	odd	2	1		700.2.e.a		2
5.b	even	2	1	inner	2800.2.g.c		2
5.c	odd	4	1		560.2.a.a		1
5.c	odd	4	1		2800.2.a.be		1
12.b	even	2	1		6300.2.k.p		2
15.e	even	4	1		5040.2.a.bd		1
20.d	odd	2	1		700.2.e.a		2
20.e	even	4	1		140.2.a.b	✓	1
20.e	even	4	1		700.2.a.b		1
28.d	even	2	1		4900.2.e.a		2
35.f	even	4	1		3920.2.a.bl		1
40.i	odd	4	1		2240.2.a.bb		1
40.k	even	4	1		2240.2.a.c		1
60.h	even	2	1		6300.2.k.p		2
60.l	odd	4	1		1260.2.a.h		1
60.l	odd	4	1		6300.2.a.bf		1
140.c	even	2	1		4900.2.e.a		2
140.j	odd	4	1		980.2.a.b		1
140.j	odd	4	1		4900.2.a.u		1
140.w	even	12	2		980.2.i.b		2
140.x	odd	12	2		980.2.i.j		2
420.w	even	4	1		8820.2.a.n		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.a.b	✓	1	20.e	even	4	1
560.2.a.a		1	5.c	odd	4	1
700.2.a.b		1	20.e	even	4	1
700.2.e.a		2	4.b	odd	2	1
700.2.e.a		2	20.d	odd	2	1
980.2.a.b		1	140.j	odd	4	1
980.2.i.b		2	140.w	even	12	2
980.2.i.j		2	140.x	odd	12	2
1260.2.a.h		1	60.l	odd	4	1
2240.2.a.c		1	40.k	even	4	1
2240.2.a.bb		1	40.i	odd	4	1
2800.2.a.be		1	5.c	odd	4	1
2800.2.g.c		2	1.a	even	1	1	trivial
2800.2.g.c		2	5.b	even	2	1	inner
3920.2.a.bl		1	35.f	even	4	1
4900.2.a.u		1	140.j	odd	4	1
4900.2.e.a		2	28.d	even	2	1
4900.2.e.a		2	140.c	even	2	1
5040.2.a.bd		1	15.e	even	4	1
6300.2.a.bf		1	60.l	odd	4	1
6300.2.k.p		2	12.b	even	2	1
6300.2.k.p		2	60.h	even	2	1
8820.2.a.n		1	420.w	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2800, [\chi])$ :

T_{3}^{2} + 9

T3^2 + 9

T_{11} - 5

T11 - 5

T_{13}^{2} + 9

T13^2 + 9

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 1$ T^2 + 1
$11$	$(T - 5)^{2}$ (T - 5)^2
$13$	$T^{2} + 9$ T^2 + 9
$17$	$T^{2} + 1$ T^2 + 1
$19$	$(T - 6)^{2}$ (T - 6)^2
$23$	$T^{2} + 36$ T^2 + 36
$29$	$(T - 9)^{2}$ (T - 9)^2
$31$	$(T - 4)^{2}$ (T - 4)^2
$37$	$T^{2} + 4$ T^2 + 4
$41$	$(T + 4)^{2}$ (T + 4)^2
$43$	$T^{2} + 100$ T^2 + 100
$47$	$T^{2} + 1$ T^2 + 1
$53$	$T^{2} + 16$ T^2 + 16
$59$	$(T + 8)^{2}$ (T + 8)^2
$61$	$(T + 8)^{2}$ (T + 8)^2
$67$	$T^{2} + 144$ T^2 + 144
$71$	$(T + 8)^{2}$ (T + 8)^2
$73$	$T^{2} + 4$ T^2 + 4
$79$	$(T - 13)^{2}$ (T - 13)^2
$83$	$T^{2} + 16$ T^2 + 16
$89$	$(T + 4)^{2}$ (T + 4)^2
$97$	$T^{2} + 169$ T^2 + 169
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