LMFDB - Newform orbit 2548.2.l.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(373,2548)]

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, 2, 4]))

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

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

chi := DirichletCharacter("2548.373");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.l (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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 4x^{2} + 3x + 9$ x^4 - x^3 + 4x^2 + 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 364)
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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{3} - 1) q^{3} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} + 1) q^{9} + ( - \beta_{3} + 5) q^{11} + ( - \beta_{2} + 2 \beta_1 - 1) q^{13} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{15}+ \cdots + ( - 5 \beta_{3} + 8) q^{99}+O(q^{100})$ q + (b3 - 1) * q^3 + (b3 - 2b2 + b1 - 1) q^5 + (-b3 + 1) * q^9 + (-b3 + 5) * q^11 + (-b2 + 2b1 - 1) q^13 + (b3 - 3b2 + b1 - 1) q^15 + (2b3 - 3b2 + 2b1 - 2) q^17 + (-b3 - 5) * q^19 + (-2b2 - 2) q^23 + (-2b2 + 3b1 - 2) * q^25 + (-2b3 - 1) q^27 + (-b3 + 2b2 - b1 + 1) q^29 + (-5b2 + 2b1 - 5) * q^31 + (5b3 - 8) q^33 - 4b1 q^37 + (-6b2 - b1 - 6) q^39 + (-2b3 + 6b2 - 2b1 + 2) q^41 + (-b2 + 3b1 - 1) q^43 + (-b3 + 3b2 - b1 + 1) q^45 + (-2b3 - 7b2 - 2b1 + 2) q^47 + (b3 - 6b2 + b1 - 1) q^51 + (3b2 - 6b1 + 3) * q^53 + (3b3 - 5b2 + 3b1 - 3) q^55 + (-5b3 + 2) q^57 + (2b3 + b2 + 2b1 - 2) * q^59 + (4b3 - 2) q^61 + (-3b3 - 5) q^65 + 13 * q^67 + 2b1 q^69 + (-4b2 + 6b1 - 4) * q^71 + (2b2 + 4b1 + 2) * q^73 + (-9b2 - b1 - 9) q^75 + 8b2 q^79 + (2b3 - 8) q^81 + (2b3 + 7) q^83 + (-12b2 + 5b1 - 12) * q^85 + (-b3 + 3b2 - b1 + 1) q^87 + (12b2 - 3b1 + 12) * q^89 + (-6b2 + 3b1 - 6) * q^93 + (-7b3 + 15b2 - 7b1 + 7) q^95 + (-13b2 + 3b1 - 13) * q^97 + (-5b3 + 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} + 3 q^{5} + 2 q^{9} + 18 q^{11} + 5 q^{15} + 4 q^{17} - 22 q^{19} - 4 q^{23} - q^{25} - 8 q^{27} - 3 q^{29} - 8 q^{31} - 22 q^{33} - 4 q^{37} - 13 q^{39} - 10 q^{41} + q^{43} - 5 q^{45}+ \cdots + 22 q^{99}+O(q^{100})$ 4 * q - 2 * q^3 + 3 * q^5 + 2 * q^9 + 18 * q^11 + 5 * q^15 + 4 * q^17 - 22 * q^19 - 4 * q^23 - q^25 - 8 * q^27 - 3 * q^29 - 8 * q^31 - 22 * q^33 - 4 * q^37 - 13 * q^39 - 10 * q^41 + q^43 - 5 * q^45 + 16 * q^47 + 11 * q^51 + 7 * q^55 - 2 * q^57 - 4 * q^59 - 26 * q^65 + 52 * q^67 + 2 * q^69 - 2 * q^71 + 8 * q^73 - 19 * q^75 - 16 * q^79 - 28 * q^81 + 32 * q^83 - 19 * q^85 - 5 * q^87 + 21 * q^89 - 9 * q^93 - 23 * q^95 - 23 * q^97 + 22 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - x^{3} + 4x^{2} + 3x + 9$ x^4 - x^3 + 4*x^2 + 3*x + 9 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12$ (-v^3 + 4v^2 - 4v - 3) / 12
$\beta_{3}$	$=$	$( \nu^{3} + 7 ) / 4$ (v^3 + 7) / 4

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} + 3\beta_{2} + \beta _1 - 1$ b3 + 3*b2 + b1 - 1
$\nu^{3}$	$=$	$4\beta_{3} - 7$ 4*b3 - 7

Display $\beta_i$ in terms of $\nu^j$

Character values

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

$n$	$197$	$885$	$1275$
$\chi(n)$	$-1 - \beta_{2}$	$\beta_{2}$	$1$

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}

373.1

1.15139	+	1.99426i
−0.651388	−	1.12824i
1.15139	−	1.99426i
−0.651388	+	1.12824i

−2.30278

−0.151388

0.262211i

2.30278

373.2

1.30278

1.65139

−

2.86029i

−1.30278

1537.1

−2.30278

−0.151388

−

0.262211i

2.30278

1537.2

1.30278

1.65139

2.86029i

−1.30278

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.g	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.l.i		4
7.b	odd	2	1		2548.2.l.k		4
7.c	even	3	1		2548.2.i.l		4
7.c	even	3	1		2548.2.k.f		4
7.d	odd	6	1		364.2.k.c	✓	4
7.d	odd	6	1		2548.2.i.j		4
13.c	even	3	1		2548.2.i.l		4
21.g	even	6	1		3276.2.z.d		4
28.f	even	6	1		1456.2.s.m		4
91.g	even	3	1	inner	2548.2.l.i		4
91.h	even	3	1		2548.2.k.f		4
91.m	odd	6	1		2548.2.l.k		4
91.m	odd	6	1		4732.2.a.k		2
91.n	odd	6	1		2548.2.i.j		4
91.p	odd	6	1		4732.2.a.j		2
91.v	odd	6	1		364.2.k.c	✓	4
91.w	even	12	2		4732.2.g.g		4
273.r	even	6	1		3276.2.z.d		4
364.ba	even	6	1		1456.2.s.m		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
364.2.k.c	✓	4	7.d	odd	6	1
364.2.k.c	✓	4	91.v	odd	6	1
1456.2.s.m		4	28.f	even	6	1
1456.2.s.m		4	364.ba	even	6	1
2548.2.i.j		4	7.d	odd	6	1
2548.2.i.j		4	91.n	odd	6	1
2548.2.i.l		4	7.c	even	3	1
2548.2.i.l		4	13.c	even	3	1
2548.2.k.f		4	7.c	even	3	1
2548.2.k.f		4	91.h	even	3	1
2548.2.l.i		4	1.a	even	1	1	trivial
2548.2.l.i		4	91.g	even	3	1	inner
2548.2.l.k		4	7.b	odd	2	1
2548.2.l.k		4	91.m	odd	6	1
3276.2.z.d		4	21.g	even	6	1
3276.2.z.d		4	273.r	even	6	1
4732.2.a.j		2	91.p	odd	6	1
4732.2.a.k		2	91.m	odd	6	1
4732.2.g.g		4	91.w	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} + T_{3} - 3

T_{5}^{4} - 3T_{5}^{3} + 10T_{5}^{2} + 3T_{5} + 1

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$(T^{2} + T - 3)^{2}$ (T^2 + T - 3)^2
$5$	$T^{4} - 3 T^{3} + \cdots + 1$ T^4 - 3T^3 + 10T^2 + 3*T + 1
$7$	$T^{4}$ T^4
$11$	$(T^{2} - 9 T + 17)^{2}$ (T^2 - 9*T + 17)^2
$13$	$T^{4} + 13T^{2} + 169$ T^4 + 13*T^2 + 169
$17$	$T^{4} - 4 T^{3} + \cdots + 81$ T^4 - 4T^3 + 25T^2 + 36*T + 81
$19$	$(T^{2} + 11 T + 27)^{2}$ (T^2 + 11*T + 27)^2
$23$	$(T^{2} + 2 T + 4)^{2}$ (T^2 + 2*T + 4)^2
$29$	$T^{4} + 3 T^{3} + \cdots + 1$ T^4 + 3T^3 + 10T^2 - 3*T + 1
$31$	$T^{4} + 8 T^{3} + \cdots + 9$ T^4 + 8T^3 + 61T^2 + 24*T + 9
$37$	$T^{4} + 4 T^{3} + \cdots + 2304$ T^4 + 4T^3 + 64T^2 - 192*T + 2304
$41$	$T^{4} + 10 T^{3} + \cdots + 144$ T^4 + 10T^3 + 88T^2 + 120*T + 144
$43$	$T^{4} - T^{3} + \cdots + 841$ T^4 - T^3 + 30T^2 + 29T + 841
$47$	$T^{4} - 16 T^{3} + \cdots + 2601$ T^4 - 16T^3 + 205T^2 - 816*T + 2601
$53$	$T^{4} + 117 T^{2} + 13689$ T^4 + 117*T^2 + 13689
$59$	$T^{4} + 4 T^{3} + \cdots + 81$ T^4 + 4T^3 + 25T^2 - 36*T + 81
$61$	$(T^{2} - 52)^{2}$ (T^2 - 52)^2
$67$	$(T - 13)^{4}$ (T - 13)^4
$71$	$T^{4} + 2 T^{3} + \cdots + 13456$ T^4 + 2T^3 + 120T^2 - 232*T + 13456
$73$	$T^{4} - 8 T^{3} + \cdots + 1296$ T^4 - 8T^3 + 100T^2 + 288*T + 1296
$79$	$(T^{2} + 8 T + 64)^{2}$ (T^2 + 8*T + 64)^2
$83$	$(T^{2} - 16 T + 51)^{2}$ (T^2 - 16*T + 51)^2
$89$	$T^{4} - 21 T^{3} + \cdots + 6561$ T^4 - 21T^3 + 360T^2 - 1701*T + 6561
$97$	$T^{4} + 23 T^{3} + \cdots + 10609$ T^4 + 23T^3 + 426T^2 + 2369*T + 10609
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