LMFDB - Newform orbit 2624.2.d.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2624,2,Mod(2049,2624)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2624.2049"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2624 = 2^{6} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2624.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,-10,0,0,0,0, 0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.9527454904$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 82)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $\beta = \sqrt{-2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{3} - 3 \beta q^{7} + q^{9} - \beta q^{11} - 4 \beta q^{17} - 3 \beta q^{19} + 6 q^{21} - 5 q^{25} + 4 \beta q^{27} + 4 \beta q^{29} - 8 q^{31} + 2 q^{33} - 8 q^{37} + (4 \beta - 3) q^{41} + \cdots - \beta q^{99} +O(q^{100}) $ q + b * q^3 - 3b q^7 + q^9 - b * q^11 - 4b q^17 - 3b q^19 + 6 * q^21 - 5 * q^25 + 4b q^27 + 4b q^29 - 8 * q^31 + 2 * q^33 - 8 * q^37 + (4b - 3) q^41 - 4 * q^43 - 5b q^47 - 11 * q^49 + 8 * q^51 + 4b q^53 + 6 * q^57 - 12 * q^59 - 2 * q^61 - 3b q^63 - 9b q^67 - 5b q^71 - 4 * q^73 - 5b q^75 - 6 * q^77 + 3b q^79 - 5 * q^81 + 12 * q^83 - 8 * q^87 - 4b q^89 - 8b q^93 + 12b q^97 - b * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{9} + 12 q^{21} - 10 q^{25} - 16 q^{31} + 4 q^{33} - 16 q^{37} - 6 q^{41} - 8 q^{43} - 22 q^{49} + 16 q^{51} + 12 q^{57} - 24 q^{59} - 4 q^{61} - 8 q^{73} - 12 q^{77} - 10 q^{81} + 24 q^{83} - 16 q^{87}+O(q^{100}) $ 2 * q + 2 * q^9 + 12 * q^21 - 10 * q^25 - 16 * q^31 + 4 * q^33 - 16 * q^37 - 6 * q^41 - 8 * q^43 - 22 * q^49 + 16 * q^51 + 12 * q^57 - 24 * q^59 - 4 * q^61 - 8 * q^73 - 12 * q^77 - 10 * q^81 + 24 * q^83 - 16 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2624\mathbb{Z}\right)^\times$.

$n$	$129$	$575$	$1477$
$\chi(n)$	$-1$	$1$	$1$

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} $

2049.1

	−	1.41421i
		1.41421i

−

1.41421i

4.24264i

1.00000

2049.2

1.41421i

−

4.24264i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2624.2.d.f		2
4.b	odd	2	1		2624.2.d.g		2
8.b	even	2	1		656.2.d.b		2
8.d	odd	2	1		82.2.b.a	✓	2
24.f	even	2	1		738.2.d.f		2
40.e	odd	2	1		2050.2.b.f		2
40.k	even	4	2		2050.2.d.h		4
41.b	even	2	1	inner	2624.2.d.f		2
164.d	odd	2	1		2624.2.d.g		2
328.c	odd	2	1		82.2.b.a	✓	2
328.g	even	2	1		656.2.d.b		2
328.k	odd	4	2		3362.2.a.l		2
984.p	even	2	1		738.2.d.f		2
1640.j	odd	2	1		2050.2.b.f		2
1640.u	even	4	2		2050.2.d.h		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
82.2.b.a	✓	2	8.d	odd	2	1
82.2.b.a	✓	2	328.c	odd	2	1
656.2.d.b		2	8.b	even	2	1
656.2.d.b		2	328.g	even	2	1
738.2.d.f		2	24.f	even	2	1
738.2.d.f		2	984.p	even	2	1
2050.2.b.f		2	40.e	odd	2	1
2050.2.b.f		2	1640.j	odd	2	1
2050.2.d.h		4	40.k	even	4	2
2050.2.d.h		4	1640.u	even	4	2
2624.2.d.f		2	1.a	even	1	1	trivial
2624.2.d.f		2	41.b	even	2	1	inner
2624.2.d.g		2	4.b	odd	2	1
2624.2.d.g		2	164.d	odd	2	1
3362.2.a.l		2	328.k	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2624, [\chi])$:

$ T_{3}^{2} + 2 $

T3^2 + 2

$ T_{7}^{2} + 18 $

T7^2 + 18

$ T_{23} $

T23

$ T_{31} + 8 $

T31 + 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 2 $ T^2 + 2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 18 $ T^2 + 18
$11$	$ T^{2} + 2 $ T^2 + 2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} + 32 $ T^2 + 32
$19$	$ T^{2} + 18 $ T^2 + 18
$23$	$ T^{2} $ T^2
$29$	$ T^{2} + 32 $ T^2 + 32
$31$	$ (T + 8)^{2} $ (T + 8)^2
$37$	$ (T + 8)^{2} $ (T + 8)^2
$41$	$ T^{2} + 6T + 41 $ T^2 + 6*T + 41
$43$	$ (T + 4)^{2} $ (T + 4)^2
$47$	$ T^{2} + 50 $ T^2 + 50
$53$	$ T^{2} + 32 $ T^2 + 32
$59$	$ (T + 12)^{2} $ (T + 12)^2
$61$	$ (T + 2)^{2} $ (T + 2)^2
$67$	$ T^{2} + 162 $ T^2 + 162
$71$	$ T^{2} + 50 $ T^2 + 50
$73$	$ (T + 4)^{2} $ (T + 4)^2
$79$	$ T^{2} + 18 $ T^2 + 18
$83$	$ (T - 12)^{2} $ (T - 12)^2
$89$	$ T^{2} + 32 $ T^2 + 32
$97$	$ T^{2} + 288 $ T^2 + 288
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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2624 = 2^{6} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2624.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta q^{3} - 3 \beta q^{7} + q^{9} - \beta q^{11} - 4 \beta q^{17} - 3 \beta q^{19} + 6 q^{21} - 5 q^{25} + 4 \beta q^{27} + 4 \beta q^{29} - 8 q^{31} + 2 q^{33} - 8 q^{37} + (4 \beta - 3) q^{41} + \cdots - \beta q^{99} +O(q^{100}) \) q + b * q^3 - 3b q^7 + q^9 - b * q^11 - 4b q^17 - 3b q^19 + 6 * q^21 - 5 * q^25 + 4b q^27 + 4b q^29 - 8 * q^31 + 2 * q^33 - 8 * q^37 + (4b - 3) q^41 - 4 * q^43 - 5b q^47 - 11 * q^49 + 8 * q^51 + 4b q^53 + 6 * q^57 - 12 * q^59 - 2 * q^61 - 3b q^63 - 9b q^67 - 5b q^71 - 4 * q^73 - 5b q^75 - 6 * q^77 + 3b q^79 - 5 * q^81 + 12 * q^83 - 8 * q^87 - 4b q^89 - 8b q^93 + 12b q^97 - b * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{9} + 12 q^{21} - 10 q^{25} - 16 q^{31} + 4 q^{33} - 16 q^{37} - 6 q^{41} - 8 q^{43} - 22 q^{49} + 16 q^{51} + 12 q^{57} - 24 q^{59} - 4 q^{61} - 8 q^{73} - 12 q^{77} - 10 q^{81} + 24 q^{83} - 16 q^{87}+O(q^{100}) \) 2 * q + 2 * q^9 + 12 * q^21 - 10 * q^25 - 16 * q^31 + 4 * q^33 - 16 * q^37 - 6 * q^41 - 8 * q^43 - 22 * q^49 + 16 * q^51 + 12 * q^57 - 24 * q^59 - 4 * q^61 - 8 * q^73 - 12 * q^77 - 10 * q^81 + 24 * q^83 - 16 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more