LMFDB - Newform orbit 2548.2.i.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(165,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2548.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.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:	$20.3458824350$
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:	$\Z[a_1, \ldots, a_{5}]$
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.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{3} + 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{9} + ( - \zeta_{6} - 3) q^{13} + (6 \zeta_{6} - 6) q^{15} + 3 q^{17} + (2 \zeta_{6} - 2) q^{19} - 6 q^{23} + (4 \zeta_{6} - 4) q^{25} + 4 q^{27}+ \cdots - 14 \zeta_{6} q^{97} +O(q^{100})$ q + 2z q^3 + 3z q^5 + (z - 1) * q^9 + (-z - 3) * q^13 + (6z - 6) q^15 + 3 * q^17 + (2z - 2) q^19 - 6 * q^23 + (4z - 4) q^25 + 4 * q^27 + (9z - 9) q^29 + (2z - 2) q^31 - 7 * q^37 + (-8z + 2) q^39 + (3z - 3) q^41 + 4z q^43 - 3 * q^45 + 6z q^47 + 6z q^51 + (9z - 9) q^53 - 4 * q^57 + (5z - 5) q^61 + (-12z + 3) q^65 - 2z q^67 - 12z q^69 + 6z q^71 + (-z + 1) * q^73 - 8 * q^75 + 4z q^79 + 11z q^81 + 12 * q^83 + 9z q^85 - 18 * q^87 + 6 * q^89 - 4 * q^93 - 6 * q^95 - 14z q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{3} + 3 q^{5} - q^{9} - 7 q^{13} - 6 q^{15} + 6 q^{17} - 2 q^{19} - 12 q^{23} - 4 q^{25} + 8 q^{27} - 9 q^{29} - 2 q^{31} - 14 q^{37} - 4 q^{39} - 3 q^{41} + 4 q^{43} - 6 q^{45} + 6 q^{47} + 6 q^{51}+ \cdots - 14 q^{97}+O(q^{100})$ 2 * q + 2 * q^3 + 3 * q^5 - q^9 - 7 * q^13 - 6 * q^15 + 6 * q^17 - 2 * q^19 - 12 * q^23 - 4 * q^25 + 8 * q^27 - 9 * q^29 - 2 * q^31 - 14 * q^37 - 4 * q^39 - 3 * q^41 + 4 * q^43 - 6 * q^45 + 6 * q^47 + 6 * q^51 - 9 * q^53 - 8 * q^57 - 5 * q^61 - 6 * q^65 - 2 * q^67 - 12 * q^69 + 6 * q^71 + q^73 - 16 * q^75 + 4 * q^79 + 11 * q^81 + 24 * q^83 + 9 * q^85 - 36 * q^87 + 12 * q^89 - 8 * q^93 - 12 * q^95 - 14 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times$ .

$n$	$197$	$885$	$1275$
$\chi(n)$	$-\zeta_{6}$	$-\zeta_{6}$	$1$

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}

165.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

1.73205i

1.50000

2.59808i

−0.500000

0.866025i

1745.1

1.00000

−

1.73205i

1.50000

−

2.59808i

−0.500000

−

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.2.i.g		2
7.b	odd	2	1		2548.2.i.b		2
7.c	even	3	1		52.2.e.b	✓	2
7.c	even	3	1		2548.2.l.b		2
7.d	odd	6	1		2548.2.k.a		2
7.d	odd	6	1		2548.2.l.g		2
13.c	even	3	1		2548.2.l.b		2
21.h	odd	6	1		468.2.l.d		2
28.g	odd	6	1		208.2.i.a		2
35.j	even	6	1		1300.2.i.b		2
35.l	odd	12	2		1300.2.bb.d		4
56.k	odd	6	1		832.2.i.i		2
56.p	even	6	1		832.2.i.c		2
84.n	even	6	1		1872.2.t.m		2
91.g	even	3	1		52.2.e.b	✓	2
91.h	even	3	1		676.2.a.a		1
91.h	even	3	1	inner	2548.2.i.g		2
91.k	even	6	1		676.2.a.b		1
91.m	odd	6	1		2548.2.k.a		2
91.n	odd	6	1		2548.2.l.g		2
91.r	even	6	1		676.2.e.d		2
91.u	even	6	1		676.2.e.d		2
91.v	odd	6	1		2548.2.i.b		2
91.x	odd	12	2		676.2.d.a		2
91.z	odd	12	2		676.2.h.d		4
91.bd	odd	12	2		676.2.h.d		4
273.s	odd	6	1		6084.2.a.o		1
273.bm	odd	6	1		468.2.l.d		2
273.bp	odd	6	1		6084.2.a.c		1
273.bv	even	12	2		6084.2.b.k		2
364.q	odd	6	1		208.2.i.a		2
364.bi	odd	6	1		2704.2.a.l		1
364.bk	odd	6	1		2704.2.a.m		1
364.ca	even	12	2		2704.2.f.i		2
455.bm	even	6	1		1300.2.i.b		2
455.cx	odd	12	2		1300.2.bb.d		4
728.bg	even	6	1		832.2.i.c		2
728.di	odd	6	1		832.2.i.i		2
1092.dc	even	6	1		1872.2.t.m		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
52.2.e.b	✓	2	7.c	even	3	1
52.2.e.b	✓	2	91.g	even	3	1
208.2.i.a		2	28.g	odd	6	1
208.2.i.a		2	364.q	odd	6	1
468.2.l.d		2	21.h	odd	6	1
468.2.l.d		2	273.bm	odd	6	1
676.2.a.a		1	91.h	even	3	1
676.2.a.b		1	91.k	even	6	1
676.2.d.a		2	91.x	odd	12	2
676.2.e.d		2	91.r	even	6	1
676.2.e.d		2	91.u	even	6	1
676.2.h.d		4	91.z	odd	12	2
676.2.h.d		4	91.bd	odd	12	2
832.2.i.c		2	56.p	even	6	1
832.2.i.c		2	728.bg	even	6	1
832.2.i.i		2	56.k	odd	6	1
832.2.i.i		2	728.di	odd	6	1
1300.2.i.b		2	35.j	even	6	1
1300.2.i.b		2	455.bm	even	6	1
1300.2.bb.d		4	35.l	odd	12	2
1300.2.bb.d		4	455.cx	odd	12	2
1872.2.t.m		2	84.n	even	6	1
1872.2.t.m		2	1092.dc	even	6	1
2548.2.i.b		2	7.b	odd	2	1
2548.2.i.b		2	91.v	odd	6	1
2548.2.i.g		2	1.a	even	1	1	trivial
2548.2.i.g		2	91.h	even	3	1	inner
2548.2.k.a		2	7.d	odd	6	1
2548.2.k.a		2	91.m	odd	6	1
2548.2.l.b		2	7.c	even	3	1
2548.2.l.b		2	13.c	even	3	1
2548.2.l.g		2	7.d	odd	6	1
2548.2.l.g		2	91.n	odd	6	1
2704.2.a.l		1	364.bi	odd	6	1
2704.2.a.m		1	364.bk	odd	6	1
2704.2.f.i		2	364.ca	even	12	2
6084.2.a.c		1	273.bp	odd	6	1
6084.2.a.o		1	273.s	odd	6	1
6084.2.b.k		2	273.bv	even	12	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2548, [\chi])$ :

T_{3}^{2} - 2T_{3} + 4

T_{5}^{2} - 3T_{5} + 9

Hecke characteristic polynomials

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