LMFDB - Newform orbit 2592.2.i.w

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(865,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("2592.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2592 = 2^{5} \cdot 3^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2592.i (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:	$20.6972242039$
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 96)
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.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + ( - 4 \zeta_{6} + 4) q^{11} + 2 \zeta_{6} q^{13} + 6 q^{17} - 4 q^{19} + ( - \zeta_{6} + 1) q^{25} + ( - 2 \zeta_{6} + 2) q^{29} - 4 \zeta_{6} q^{31} + \cdots + ( - 14 \zeta_{6} + 14) q^{97} +O(q^{100})$ q + 2z q^5 + (-4z + 4) q^7 + (-4z + 4) q^11 + 2z q^13 + 6 * q^17 - 4 * q^19 + (-z + 1) * q^25 + (-2z + 2) q^29 - 4z q^31 + 8 * q^35 - 2 * q^37 + 2z q^41 + (4z - 4) q^43 + (-8z + 8) q^47 - 9z q^49 - 10 * q^53 + 8 * q^55 - 4z q^59 + (6z - 6) q^61 + (4z - 4) q^65 - 4z q^67 + 16 * q^71 - 6 * q^73 - 16z q^77 + (4z - 4) q^79 + (-12z + 12) q^83 + 12z q^85 - 10 * q^89 + 8 * q^91 - 8z q^95 + (-14z + 14) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 12 q^{17} - 8 q^{19} + q^{25} + 2 q^{29} - 4 q^{31} + 16 q^{35} - 4 q^{37} + 2 q^{41} - 4 q^{43} + 8 q^{47} - 9 q^{49} - 20 q^{53} + 16 q^{55} - 4 q^{59}+ \cdots + 14 q^{97}+O(q^{100})$ 2 * q + 2 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 12 * q^17 - 8 * q^19 + q^25 + 2 * q^29 - 4 * q^31 + 16 * q^35 - 4 * q^37 + 2 * q^41 - 4 * q^43 + 8 * q^47 - 9 * q^49 - 20 * q^53 + 16 * q^55 - 4 * q^59 - 6 * q^61 - 4 * q^65 - 4 * q^67 + 32 * q^71 - 12 * q^73 - 16 * q^77 - 4 * q^79 + 12 * q^83 + 12 * q^85 - 20 * q^89 + 16 * q^91 - 8 * q^95 + 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$ .

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-\zeta_{6}$	$1$

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}

865.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

1.73205i

2.00000

−

3.46410i

1729.1

1.00000

−

1.73205i

2.00000

3.46410i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2592.2.i.w		2
3.b	odd	2	1		2592.2.i.h		2
4.b	odd	2	1		2592.2.i.q		2
9.c	even	3	1		288.2.a.b		1
9.c	even	3	1	inner	2592.2.i.w		2
9.d	odd	6	1		96.2.a.b	yes	1
9.d	odd	6	1		2592.2.i.h		2
12.b	even	2	1		2592.2.i.b		2
36.f	odd	6	1		288.2.a.c		1
36.f	odd	6	1		2592.2.i.q		2
36.h	even	6	1		96.2.a.a	✓	1
36.h	even	6	1		2592.2.i.b		2
45.h	odd	6	1		2400.2.a.q		1
45.j	even	6	1		7200.2.a.bx		1
45.k	odd	12	2		7200.2.f.f		2
45.l	even	12	2		2400.2.f.r		2
63.o	even	6	1		4704.2.a.e		1
72.j	odd	6	1		192.2.a.a		1
72.l	even	6	1		192.2.a.c		1
72.n	even	6	1		576.2.a.g		1
72.p	odd	6	1		576.2.a.h		1
144.u	even	12	2		768.2.d.a		2
144.v	odd	12	2		2304.2.d.c		2
144.w	odd	12	2		768.2.d.h		2
144.x	even	12	2		2304.2.d.s		2
180.n	even	6	1		2400.2.a.r		1
180.p	odd	6	1		7200.2.a.e		1
180.v	odd	12	2		2400.2.f.a		2
180.x	even	12	2		7200.2.f.x		2
252.s	odd	6	1		4704.2.a.t		1
360.bd	even	6	1		4800.2.a.f		1
360.bh	odd	6	1		4800.2.a.co		1
360.br	even	12	2		4800.2.f.e		2
360.bt	odd	12	2		4800.2.f.bh		2
504.cc	even	6	1		9408.2.a.ct		1
504.co	odd	6	1		9408.2.a.bj		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
96.2.a.a	✓	1	36.h	even	6	1
96.2.a.b	yes	1	9.d	odd	6	1
192.2.a.a		1	72.j	odd	6	1
192.2.a.c		1	72.l	even	6	1
288.2.a.b		1	9.c	even	3	1
288.2.a.c		1	36.f	odd	6	1
576.2.a.g		1	72.n	even	6	1
576.2.a.h		1	72.p	odd	6	1
768.2.d.a		2	144.u	even	12	2
768.2.d.h		2	144.w	odd	12	2
2304.2.d.c		2	144.v	odd	12	2
2304.2.d.s		2	144.x	even	12	2
2400.2.a.q		1	45.h	odd	6	1
2400.2.a.r		1	180.n	even	6	1
2400.2.f.a		2	180.v	odd	12	2
2400.2.f.r		2	45.l	even	12	2
2592.2.i.b		2	12.b	even	2	1
2592.2.i.b		2	36.h	even	6	1
2592.2.i.h		2	3.b	odd	2	1
2592.2.i.h		2	9.d	odd	6	1
2592.2.i.q		2	4.b	odd	2	1
2592.2.i.q		2	36.f	odd	6	1
2592.2.i.w		2	1.a	even	1	1	trivial
2592.2.i.w		2	9.c	even	3	1	inner
4704.2.a.e		1	63.o	even	6	1
4704.2.a.t		1	252.s	odd	6	1
4800.2.a.f		1	360.bd	even	6	1
4800.2.a.co		1	360.bh	odd	6	1
4800.2.f.e		2	360.br	even	12	2
4800.2.f.bh		2	360.bt	odd	12	2
7200.2.a.e		1	180.p	odd	6	1
7200.2.a.bx		1	45.j	even	6	1
7200.2.f.f		2	45.k	odd	12	2
7200.2.f.x		2	180.x	even	12	2
9408.2.a.bj		1	504.co	odd	6	1
9408.2.a.ct		1	504.cc	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2592, [\chi])$ :

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

T_{7}^{2} - 4T_{7} + 16

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

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
$7$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$11$	$T^{2} - 4T + 16$ T^2 - 4*T + 16
$13$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$17$	$(T - 6)^{2}$ (T - 6)^2
$19$	$(T + 4)^{2}$ (T + 4)^2
$23$	$T^{2}$ T^2
$29$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$31$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$37$	$(T + 2)^{2}$ (T + 2)^2
$41$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$43$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$47$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$53$	$(T + 10)^{2}$ (T + 10)^2
$59$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$61$	$T^{2} + 6T + 36$ T^2 + 6*T + 36
$67$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$71$	$(T - 16)^{2}$ (T - 16)^2
$73$	$(T + 6)^{2}$ (T + 6)^2
$79$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$83$	$T^{2} - 12T + 144$ T^2 - 12*T + 144
$89$	$(T + 10)^{2}$ (T + 10)^2
$97$	$T^{2} - 14T + 196$ T^2 - 14*T + 196
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