LMFDB - Newform orbit 1872.2.t.m

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,2,Mod(289,1872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1872, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("1872.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1872 = 2^{4} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1872.t (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:	$14.9479952584$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 52)
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.

$f(q)$	$=$	$q + 3 q^{5} - 4 \zeta_{6} q^{7} + ( - \zeta_{6} - 3) q^{13} + 3 \zeta_{6} q^{17} + 2 \zeta_{6} q^{19} + ( - 6 \zeta_{6} + 6) q^{23} + 4 q^{25} + ( - 9 \zeta_{6} + 9) q^{29} - 2 q^{31} - 12 \zeta_{6} q^{35} + \cdots - 14 \zeta_{6} q^{97} +O(q^{100})$ q + 3 * q^5 - 4z q^7 + (-z - 3) * q^13 + 3z q^17 + 2z q^19 + (-6z + 6) q^23 + 4 * q^25 + (-9z + 9) q^29 - 2 * q^31 - 12z q^35 + (-7z + 7) q^37 + (-3z + 3) q^41 - 4z q^43 - 6 * q^47 + (9z - 9) q^49 - 9 * q^53 - 5z q^61 + (-3z - 9) q^65 + (-2z + 2) q^67 + 6z q^71 - q^73 + 4 * q^79 + 12 * q^83 + 9z q^85 + (-6z + 6) q^89 + (16z - 4) q^91 + 6z q^95 - 14z q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 6 q^{5} - 4 q^{7} - 7 q^{13} + 3 q^{17} + 2 q^{19} + 6 q^{23} + 8 q^{25} + 9 q^{29} - 4 q^{31} - 12 q^{35} + 7 q^{37} + 3 q^{41} - 4 q^{43} - 12 q^{47} - 9 q^{49} - 18 q^{53} - 5 q^{61} - 21 q^{65}+ \cdots - 14 q^{97}+O(q^{100})$ 2 * q + 6 * q^5 - 4 * q^7 - 7 * q^13 + 3 * q^17 + 2 * q^19 + 6 * q^23 + 8 * q^25 + 9 * q^29 - 4 * q^31 - 12 * q^35 + 7 * q^37 + 3 * q^41 - 4 * q^43 - 12 * q^47 - 9 * q^49 - 18 * q^53 - 5 * q^61 - 21 * q^65 + 2 * q^67 + 6 * q^71 - 2 * q^73 + 8 * q^79 + 24 * q^83 + 9 * q^85 + 6 * q^89 + 8 * q^91 + 6 * q^95 - 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$ .

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-\zeta_{6}$	$1$	$1$	$1$

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

0.500000	−	0.866025i
0.500000	+	0.866025i

3.00000

−2.00000

3.46410i

1153.1

3.00000

−2.00000

−

3.46410i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1872.2.t.m		2
3.b	odd	2	1		208.2.i.a		2
4.b	odd	2	1		468.2.l.d		2
12.b	even	2	1		52.2.e.b	✓	2
13.c	even	3	1	inner	1872.2.t.m		2
24.f	even	2	1		832.2.i.c		2
24.h	odd	2	1		832.2.i.i		2
39.h	odd	6	1		2704.2.a.m		1
39.i	odd	6	1		208.2.i.a		2
39.i	odd	6	1		2704.2.a.l		1
39.k	even	12	2		2704.2.f.i		2
52.i	odd	6	1		6084.2.a.c		1
52.j	odd	6	1		468.2.l.d		2
52.j	odd	6	1		6084.2.a.o		1
52.l	even	12	2		6084.2.b.k		2
60.h	even	2	1		1300.2.i.b		2
60.l	odd	4	2		1300.2.bb.d		4
84.h	odd	2	1		2548.2.k.a		2
84.j	odd	6	1		2548.2.i.b		2
84.j	odd	6	1		2548.2.l.g		2
84.n	even	6	1		2548.2.i.g		2
84.n	even	6	1		2548.2.l.b		2
156.h	even	2	1		676.2.e.d		2
156.l	odd	4	2		676.2.h.d		4
156.p	even	6	1		52.2.e.b	✓	2
156.p	even	6	1		676.2.a.a		1
156.r	even	6	1		676.2.a.b		1
156.r	even	6	1		676.2.e.d		2
156.v	odd	12	2		676.2.d.a		2
156.v	odd	12	2		676.2.h.d		4
312.bh	odd	6	1		832.2.i.i		2
312.bn	even	6	1		832.2.i.c		2
780.br	even	6	1		1300.2.i.b		2
780.cj	odd	12	2		1300.2.bb.d		4
1092.bt	odd	6	1		2548.2.l.g		2
1092.cc	odd	6	1		2548.2.i.b		2
1092.ck	even	6	1		2548.2.l.b		2
1092.dc	even	6	1		2548.2.i.g		2
1092.dd	odd	6	1		2548.2.k.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.e.b	✓	2	12.b	even	2	1
52.2.e.b	✓	2	156.p	even	6	1
208.2.i.a		2	3.b	odd	2	1
208.2.i.a		2	39.i	odd	6	1
468.2.l.d		2	4.b	odd	2	1
468.2.l.d		2	52.j	odd	6	1
676.2.a.a		1	156.p	even	6	1
676.2.a.b		1	156.r	even	6	1
676.2.d.a		2	156.v	odd	12	2
676.2.e.d		2	156.h	even	2	1
676.2.e.d		2	156.r	even	6	1
676.2.h.d		4	156.l	odd	4	2
676.2.h.d		4	156.v	odd	12	2
832.2.i.c		2	24.f	even	2	1
832.2.i.c		2	312.bn	even	6	1
832.2.i.i		2	24.h	odd	2	1
832.2.i.i		2	312.bh	odd	6	1
1300.2.i.b		2	60.h	even	2	1
1300.2.i.b		2	780.br	even	6	1
1300.2.bb.d		4	60.l	odd	4	2
1300.2.bb.d		4	780.cj	odd	12	2
1872.2.t.m		2	1.a	even	1	1	trivial
1872.2.t.m		2	13.c	even	3	1	inner
2548.2.i.b		2	84.j	odd	6	1
2548.2.i.b		2	1092.cc	odd	6	1
2548.2.i.g		2	84.n	even	6	1
2548.2.i.g		2	1092.dc	even	6	1
2548.2.k.a		2	84.h	odd	2	1
2548.2.k.a		2	1092.dd	odd	6	1
2548.2.l.b		2	84.n	even	6	1
2548.2.l.b		2	1092.ck	even	6	1
2548.2.l.g		2	84.j	odd	6	1
2548.2.l.g		2	1092.bt	odd	6	1
2704.2.a.l		1	39.i	odd	6	1
2704.2.a.m		1	39.h	odd	6	1
2704.2.f.i		2	39.k	even	12	2
6084.2.a.c		1	52.i	odd	6	1
6084.2.a.o		1	52.j	odd	6	1
6084.2.b.k		2	52.l	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1872, [\chi])$ :

T_{5} - 3

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

T_{11}

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T - 3)^{2}$ (T - 3)^2
$7$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$11$	$T^{2}$ T^2
$13$	$T^{2} + 7T + 13$ T^2 + 7*T + 13
$17$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$19$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$23$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$29$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$31$	$(T + 2)^{2}$ (T + 2)^2
$37$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$41$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$43$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$47$	$(T + 6)^{2}$ (T + 6)^2
$53$	$(T + 9)^{2}$ (T + 9)^2
$59$	$T^{2}$ T^2
$61$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$67$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$71$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$73$	$(T + 1)^{2}$ (T + 1)^2
$79$	$(T - 4)^{2}$ (T - 4)^2
$83$	$(T - 12)^{2}$ (T - 12)^2
$89$	$T^{2} - 6T + 36$ T^2 - 6*T + 36
$97$	$T^{2} + 14T + 196$ T^2 + 14*T + 196
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