LMFDB - Newform orbit 3528.2.s.y

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3528 = 2^{3} \cdot 3^{2} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$ , degree $2$ , 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:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -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 a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{11} + 2 q^{13} + (2 \zeta_{6} - 2) q^{17} - 4 \zeta_{6} q^{19} - 8 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - 6 q^{29} + ( - 8 \zeta_{6} + 8) q^{31} + \cdots - 2 q^{97} +O(q^{100})$ q + 2z q^5 + (-4z + 4) q^11 + 2 * q^13 + (2z - 2) q^17 - 4z q^19 - 8z q^23 + (-z + 1) * q^25 - 6 * q^29 + (-8z + 8) q^31 - 6z q^37 - 6 * q^41 + 4 * q^43 + (2z - 2) q^53 + 8 * q^55 + (4z - 4) q^59 - 2z q^61 + 4z q^65 + (-4z + 4) q^67 - 8 * q^71 + (-10z + 10) q^73 + 8z q^79 - 4 * q^83 - 4 * q^85 + 6z q^89 + (-8z + 8) q^95 - 2 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 4 q^{11} + 4 q^{13} - 2 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} - 12 q^{29} + 8 q^{31} - 6 q^{37} - 12 q^{41} + 8 q^{43} - 2 q^{53} + 16 q^{55} - 4 q^{59} - 2 q^{61} + 4 q^{65} + 4 q^{67}+ \cdots - 4 q^{97}+O(q^{100})$ 2 * q + 2 * q^5 + 4 * q^11 + 4 * q^13 - 2 * q^17 - 4 * q^19 - 8 * q^23 + q^25 - 12 * q^29 + 8 * q^31 - 6 * q^37 - 12 * q^41 + 8 * q^43 - 2 * q^53 + 16 * q^55 - 4 * q^59 - 2 * q^61 + 4 * q^65 + 4 * q^67 - 16 * q^71 + 10 * q^73 + 8 * q^79 - 8 * q^83 - 8 * q^85 + 6 * q^89 + 8 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$-\zeta_{6}$	$1$	$1$

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}

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

1.73205i

3313.1

1.00000

−

1.73205i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3528.2.s.y		2
3.b	odd	2	1		1176.2.q.a		2
7.b	odd	2	1		3528.2.s.j		2
7.c	even	3	1		3528.2.a.d		1
7.c	even	3	1	inner	3528.2.s.y		2
7.d	odd	6	1		72.2.a.a		1
7.d	odd	6	1		3528.2.s.j		2
12.b	even	2	1		2352.2.q.r		2
21.c	even	2	1		1176.2.q.i		2
21.g	even	6	1		24.2.a.a	✓	1
21.g	even	6	1		1176.2.q.i		2
21.h	odd	6	1		1176.2.a.i		1
21.h	odd	6	1		1176.2.q.a		2
28.f	even	6	1		144.2.a.b		1
28.g	odd	6	1		7056.2.a.q		1
35.i	odd	6	1		1800.2.a.m		1
35.k	even	12	2		1800.2.f.c		2
56.j	odd	6	1		576.2.a.d		1
56.m	even	6	1		576.2.a.b		1
63.i	even	6	1		648.2.i.g		2
63.k	odd	6	1		648.2.i.b		2
63.s	even	6	1		648.2.i.g		2
63.t	odd	6	1		648.2.i.b		2
77.i	even	6	1		8712.2.a.u		1
84.h	odd	2	1		2352.2.q.l		2
84.j	odd	6	1		48.2.a.a		1
84.j	odd	6	1		2352.2.q.l		2
84.n	even	6	1		2352.2.a.i		1
84.n	even	6	1		2352.2.q.r		2
105.p	even	6	1		600.2.a.h		1
105.w	odd	12	2		600.2.f.e		2
112.v	even	12	2		2304.2.d.k		2
112.x	odd	12	2		2304.2.d.i		2
140.s	even	6	1		3600.2.a.v		1
140.x	odd	12	2		3600.2.f.r		2
168.s	odd	6	1		9408.2.a.h		1
168.v	even	6	1		9408.2.a.cc		1
168.ba	even	6	1		192.2.a.d		1
168.be	odd	6	1		192.2.a.b		1
231.k	odd	6	1		2904.2.a.c		1
252.n	even	6	1		1296.2.i.e		2
252.r	odd	6	1		1296.2.i.m		2
252.bj	even	6	1		1296.2.i.e		2
252.bn	odd	6	1		1296.2.i.m		2
273.ba	even	6	1		4056.2.a.i		1
273.cb	odd	12	2		4056.2.c.e		2
336.bo	even	12	2		768.2.d.e		2
336.br	odd	12	2		768.2.d.d		2
357.s	even	6	1		6936.2.a.p		1
399.s	odd	6	1		8664.2.a.j		1
420.be	odd	6	1		1200.2.a.d		1
420.br	even	12	2		1200.2.f.b		2
840.cb	even	6	1		4800.2.a.q		1
840.ct	odd	6	1		4800.2.a.cc		1
840.dh	odd	12	2		4800.2.f.d		2
840.dk	even	12	2		4800.2.f.bg		2
924.y	even	6	1		5808.2.a.s		1
1092.ct	odd	6	1		8112.2.a.be		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
24.2.a.a	✓	1	21.g	even	6	1
48.2.a.a		1	84.j	odd	6	1
72.2.a.a		1	7.d	odd	6	1
144.2.a.b		1	28.f	even	6	1
192.2.a.b		1	168.be	odd	6	1
192.2.a.d		1	168.ba	even	6	1
576.2.a.b		1	56.m	even	6	1
576.2.a.d		1	56.j	odd	6	1
600.2.a.h		1	105.p	even	6	1
600.2.f.e		2	105.w	odd	12	2
648.2.i.b		2	63.k	odd	6	1
648.2.i.b		2	63.t	odd	6	1
648.2.i.g		2	63.i	even	6	1
648.2.i.g		2	63.s	even	6	1
768.2.d.d		2	336.br	odd	12	2
768.2.d.e		2	336.bo	even	12	2
1176.2.a.i		1	21.h	odd	6	1
1176.2.q.a		2	3.b	odd	2	1
1176.2.q.a		2	21.h	odd	6	1
1176.2.q.i		2	21.c	even	2	1
1176.2.q.i		2	21.g	even	6	1
1200.2.a.d		1	420.be	odd	6	1
1200.2.f.b		2	420.br	even	12	2
1296.2.i.e		2	252.n	even	6	1
1296.2.i.e		2	252.bj	even	6	1
1296.2.i.m		2	252.r	odd	6	1
1296.2.i.m		2	252.bn	odd	6	1
1800.2.a.m		1	35.i	odd	6	1
1800.2.f.c		2	35.k	even	12	2
2304.2.d.i		2	112.x	odd	12	2
2304.2.d.k		2	112.v	even	12	2
2352.2.a.i		1	84.n	even	6	1
2352.2.q.l		2	84.h	odd	2	1
2352.2.q.l		2	84.j	odd	6	1
2352.2.q.r		2	12.b	even	2	1
2352.2.q.r		2	84.n	even	6	1
2904.2.a.c		1	231.k	odd	6	1
3528.2.a.d		1	7.c	even	3	1
3528.2.s.j		2	7.b	odd	2	1
3528.2.s.j		2	7.d	odd	6	1
3528.2.s.y		2	1.a	even	1	1	trivial
3528.2.s.y		2	7.c	even	3	1	inner
3600.2.a.v		1	140.s	even	6	1
3600.2.f.r		2	140.x	odd	12	2
4056.2.a.i		1	273.ba	even	6	1
4056.2.c.e		2	273.cb	odd	12	2
4800.2.a.q		1	840.cb	even	6	1
4800.2.a.cc		1	840.ct	odd	6	1
4800.2.f.d		2	840.dh	odd	12	2
4800.2.f.bg		2	840.dk	even	12	2
5808.2.a.s		1	924.y	even	6	1
6936.2.a.p		1	357.s	even	6	1
7056.2.a.q		1	28.g	odd	6	1
8112.2.a.be		1	1092.ct	odd	6	1
8664.2.a.j		1	399.s	odd	6	1
8712.2.a.u		1	77.i	even	6	1
9408.2.a.h		1	168.s	odd	6	1
9408.2.a.cc		1	168.v	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(3528, [\chi])$ :

T_{5}^{2} - 2T_{5} + 4

T_{11}^{2} - 4T_{11} + 16

T_{13} - 2

T_{23}^{2} + 8T_{23} + 64

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$7$	$T^{2}$ T^2
$11$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$13$	$(T - 2)^{2}$ (T - 2)^2
$17$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$19$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$23$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$29$	$(T + 6)^{2}$ (T + 6)^2
$31$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$37$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$41$	$(T + 6)^{2}$ (T + 6)^2
$43$	$(T - 4)^{2}$ (T - 4)^2
$47$	$T^{2}$ T^2
$53$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$59$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$61$	$T^{2} + 2T + 4$ T^2 + 2*T + 4
$67$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$71$	$(T + 8)^{2}$ (T + 8)^2
$73$	$T^{2} - 10T + 100$ T^2 - 10*T + 100
$79$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$83$	$(T + 4)^{2}$ (T + 4)^2
$89$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$97$	$(T + 2)^{2}$ (T + 2)^2
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