LMFDB - Newform orbit 2548.2.a.r

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2548,2,Mod(1,2548)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2548, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2548.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2548 = 2^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2548.a (trivial)

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:	yes
Analytic conductor:	$20.3458824350$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.39110656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 10x^{4} - 4x^{3} + 20x^{2} + 16x + 2$ x^6 - 10x^4 - 4x^3 + 20x^2 + 16x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + (\beta_{4} + \beta_1) q^{3} + (\beta_{3} - 2) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{9} + (\beta_{3} - 2 \beta_1 - 1) q^{11} + q^{13} + (\beta_{5} - 3 \beta_{4} - \beta_{3} + \cdots + 1) q^{15}+ \cdots + (\beta_{5} - 8 \beta_{4} + \beta_{3} + \cdots - 8) q^{99}+O(q^{100})$ q + (b4 + b1) * q^3 + (b3 - 2) * q^5 + (-b3 - b2 + b1 + 1) * q^9 + (b3 - 2b1 - 1) q^11 + q^13 + (b5 - 3b4 - b3 + b2 - 2b1 + 1) * q^15 + (-b5 + b2 - b1 - 2) * q^17 + (b5 - 2b4 - 2b3 + b2 + b1 - 1) * q^19 + (-2b5 + b3 - b1) q^23 + (-3b3 - b1 + 3) q^25 + (-2b5 + 3b4 - 2b2 + 2) q^27 + (-2b4 - 2b3 + 1) * q^29 + (b5 - b4 - b3 - 1) * q^31 + (b5 - 2b4 + b3 + b2 - 3b1 - 5) * q^33 + (2b5 - 2b4 - b3 + 2b2 + 3) q^37 + (b4 + b1) * q^39 + (b5 - b4 + 2b3 - 2b2 - 2b1 - 3) q^41 + (-2b5 - 2b2 + 1) * q^43 + (-b5 + 2b3 + 3b2 - 3b1 - 5) q^45 + (b5 - 2b4 + b2 + b1 - 5) q^47 + (2b5 - 8b4 - b3 + b2 - 3b1 - 2) q^51 + (2b4 + 3b3 - 3b1 - 2) q^53 + (2b4 - 2b2 + 3b1 + 2) q^55 + (-2b5 - b4 + 3b3 + b2 - 2b1 - 2) q^57 + (-b4 - b3 - 3b2 + b1 - 2) q^59 + (-b5 + 2b4 - 2b3 - b2 - 2) * q^61 + (b3 - 2) * q^65 + (b5 + 3b4 + b2 - 2b1) * q^67 + (3b5 - 5b4 - 4b3 + b2 + b1) q^69 + (b5 + 2b4 + 3b3 - 2b2 - 2) q^71 + (3b5 + 2b4 + b2 + 3b1 - 3) q^73 + (-3b5 + 6b4 + 4b3 - 3b2 + 2b1 - 6) q^75 + (-4b4 - b2 + 2) q^79 + (6b4 - b3 - 2b2 + 3b1 + 2) q^81 + (-2b4 - 3b3 + 2b2 - 2b1 + 4) * q^83 + (2b5 - 4b4 - 3b3 - 4b2 + 4b1 + 3) q^85 + (-2b5 + 3b4 + 2b3 + b1 - 4) q^87 + (-2b5 + 5b4 - 2b3 - b2 + 3b1 - 4) * q^89 + (-2b5 + 2b4 + 3b3 - 2b1 - 3) * q^93 + (-2b5 + 6b4 + 2b1 - 1) q^95 + (-2b5 - b4 - 3b2 - b1 - 4) * q^97 + (b5 - 8b4 + b3 + 4b2 - 4b1 - 8) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 10 q^{5} + 6 q^{9} - 4 q^{11} + 6 q^{13} + 4 q^{15} - 16 q^{17} - 10 q^{19} - 2 q^{23} + 12 q^{25} + 12 q^{27} + 2 q^{29} - 6 q^{31} - 28 q^{33} + 16 q^{37} - 8 q^{41} + 6 q^{43} - 34 q^{45} - 30 q^{47}+ \cdots - 52 q^{99}+O(q^{100})$ 6 * q - 10 * q^5 + 6 * q^9 - 4 * q^11 + 6 * q^13 + 4 * q^15 - 16 * q^17 - 10 * q^19 - 2 * q^23 + 12 * q^25 + 12 * q^27 + 2 * q^29 - 6 * q^31 - 28 * q^33 + 16 * q^37 - 8 * q^41 + 6 * q^43 - 34 * q^45 - 30 * q^47 - 12 * q^51 - 6 * q^53 + 16 * q^55 - 12 * q^57 - 8 * q^59 - 16 * q^61 - 10 * q^65 - 4 * q^69 - 14 * q^73 - 28 * q^75 + 14 * q^79 + 14 * q^81 + 14 * q^83 + 24 * q^85 - 24 * q^87 - 30 * q^89 - 16 * q^93 - 10 * q^95 - 22 * q^97 - 52 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 10x^{4} - 4x^{3} + 20x^{2} + 16x + 2$ x^6 - 10*x^4 - 4*x^3 + 20*x^2 + 16*x + 2 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -2\nu^{5} + 3\nu^{4} + 18\nu^{3} - 14\nu^{2} - 34\nu - 11 ) / 5$ (-2v^5 + 3v^4 + 18v^3 - 14v^2 - 34*v - 11) / 5
$\beta_{3}$	$=$	$( -2\nu^{5} + 3\nu^{4} + 18\nu^{3} - 19\nu^{2} - 29\nu + 9 ) / 5$ (-2v^5 + 3v^4 + 18v^3 - 19v^2 - 29*v + 9) / 5
$\beta_{4}$	$=$	$( -3\nu^{5} + 2\nu^{4} + 27\nu^{3} - 6\nu^{2} - 46\nu - 14 ) / 5$ (-3v^5 + 2v^4 + 27v^3 - 6v^2 - 46*v - 14) / 5
$\beta_{5}$	$=$	$( -4\nu^{5} + \nu^{4} + 41\nu^{3} + 2\nu^{2} - 88\nu - 27 ) / 5$ (-4v^5 + v^4 + 41v^3 + 2v^2 - 88v - 27) / 5

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{3} + \beta_{2} + \beta _1 + 4$ -b3 + b2 + b1 + 4
$\nu^{3}$	$=$	$\beta_{5} - 2\beta_{4} + \beta_{2} + 6\beta _1 + 2$ b5 - 2b4 + b2 + 6b1 + 2
$\nu^{4}$	$=$	$-2\beta_{4} - 6\beta_{3} + 9\beta_{2} + 8\beta _1 + 25$ -2b4 - 6b3 + 9b2 + 8b1 + 25
$\nu^{5}$	$=$	$9\beta_{5} - 21\beta_{4} - 2\beta_{3} + 13\beta_{2} + 42\beta _1 + 22$ 9b5 - 21b4 - 2b3 + 13b2 + 42*b1 + 22

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

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}

1.1

−1.30569
−0.156000
−2.47334
−0.759171
2.87591
1.81830

−2.71991

−3.85705

4.39789

1.2

−1.57021

0.599047

−0.534430

1.3

−1.05913

−4.09294

−1.87824

1.4

0.655042

0.738118

−2.57092

1.5

1.46169

−0.327782

−0.863457

1.6

3.23252

−3.05939

7.44916

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$+1$
$13$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2548.2.a.r	✓	6
7.b	odd	2	1		2548.2.a.s	yes	6
7.c	even	3	2		2548.2.j.s		12
7.d	odd	6	2		2548.2.j.r		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2548.2.a.r	✓	6	1.a	even	1	1	trivial
2548.2.a.s	yes	6	7.b	odd	2	1
2548.2.j.r		12	7.d	odd	6	2
2548.2.j.s		12	7.c	even	3	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(2548))$ :

T_{3}^{6} - 12T_{3}^{4} - 4T_{3}^{3} + 28T_{3}^{2} + 8T_{3} - 14

T_{5}^{6} + 10T_{5}^{5} + 29T_{5}^{4} + 8T_{5}^{3} - 47T_{5}^{2} + 6T_{5} + 7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 12 T^{4} + \cdots - 14$ T^6 - 12T^4 - 4T^3 + 28T^2 + 8T - 14
$5$	$T^{6} + 10 T^{5} + \cdots + 7$ T^6 + 10T^5 + 29T^4 + 8T^3 - 47T^2 + 6*T + 7
$7$	$T^{6}$ T^6
$11$	$T^{6} + 4 T^{5} + \cdots - 82$ T^6 + 4T^5 - 30T^4 - 52T^3 + 322T^2 - 252*T - 82
$13$	$(T - 1)^{6}$ (T - 1)^6
$17$	$T^{6} + 16 T^{5} + \cdots - 254$ T^6 + 16T^5 + 48T^4 - 332T^3 - 1768T^2 - 1288*T - 254
$19$	$T^{6} + 10 T^{5} + \cdots + 3647$ T^6 + 10T^5 - 17T^4 - 352T^3 - 349T^2 + 2518*T + 3647
$23$	$T^{6} + 2 T^{5} + \cdots - 4879$ T^6 + 2T^5 - 93T^4 - 152T^3 + 1919T^2 + 934*T - 4879
$29$	$T^{6} - 2 T^{5} + \cdots + 25$ T^6 - 2T^5 - 57T^4 + 100T^3 + 127T^2 - 130*T + 25
$31$	$T^{6} + 6 T^{5} + \cdots - 89$ T^6 + 6T^5 - 19T^4 - 204T^3 - 499T^2 - 442*T - 89
$37$	$T^{6} - 16 T^{5} + \cdots + 53918$ T^6 - 16T^5 - 10T^4 + 1116T^3 - 2934T^2 - 16844*T + 53918
$41$	$T^{6} + 8 T^{5} + \cdots - 76816$ T^6 + 8T^5 - 152T^4 - 712T^3 + 7800T^2 + 7200*T - 76816
$43$	$T^{6} - 6 T^{5} + \cdots + 73$ T^6 - 6T^5 - 105T^4 + 460T^3 + 2047T^2 + 1114*T + 73
$47$	$T^{6} + 30 T^{5} + \cdots - 14481$ T^6 + 30T^5 + 319T^4 + 1320T^3 + 587T^2 - 8574*T - 14481
$53$	$T^{6} + 6 T^{5} + \cdots - 32719$ T^6 + 6T^5 - 141T^4 - 968T^3 + 2751T^2 + 15906*T - 32719
$59$	$T^{6} + 8 T^{5} + \cdots - 19204$ T^6 + 8T^5 - 182T^4 - 1372T^3 + 8204T^2 + 57336*T - 19204
$61$	$T^{6} + 16 T^{5} + \cdots + 34552$ T^6 + 16T^5 - 22T^4 - 1168T^3 - 1988T^2 + 18336*T + 34552
$67$	$T^{6} - 176 T^{4} + \cdots + 19936$ T^6 - 176T^4 - 352T^3 + 7808T^2 + 30208T + 19936
$71$	$T^{6} - 212 T^{4} + \cdots - 15842$ T^6 - 212T^4 - 136T^3 + 6754T^2 - 7476T - 15842
$73$	$T^{6} + 14 T^{5} + \cdots - 1377$ T^6 + 14T^5 - 161T^4 - 2088T^3 + 1051T^2 + 6210*T - 1377
$79$	$T^{6} - 14 T^{5} + \cdots - 839$ T^6 - 14T^5 - 33T^4 + 820T^3 + 23T^2 - 9830*T - 839
$83$	$T^{6} - 14 T^{5} + \cdots - 38089$ T^6 - 14T^5 - 139T^4 + 984T^3 + 7305T^2 - 1666*T - 38089
$89$	$T^{6} + 30 T^{5} + \cdots + 271111$ T^6 + 30T^5 + 35T^4 - 6396T^3 - 62845T^2 - 126274*T + 271111
$97$	$T^{6} + 22 T^{5} + \cdots - 233$ T^6 + 22T^5 + 11T^4 - 1732T^3 - 2317T^2 + 33526*T - 233
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.6
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more