LMFDB - Newform orbit 2624.2.d.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2624,2,Mod(2049,2624)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$20.9527454904$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.25088.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 2 $ x^4 + 6*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
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 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_1 q^{3} + ( - \beta_{2} + 1) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10}) $ q + b1 * q^3 + (-b2 + 1) * q^5 + b3 * q^7 + b2 * q^9	\( q + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{9} + (\beta_{3} + 2 \beta_1) q^{11} - 2 \beta_{3} q^{13} + ( - \beta_{3} + 3 \beta_1) q^{15} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{3} - 3 \beta_1) q^{19} + ( - \beta_{2} + 1) q^{21} + (2 \beta_{2} - 2) q^{23} + ( - 2 \beta_{2} + 3) q^{25} + (\beta_{3} + \beta_1) q^{27} - 2 \beta_1 q^{29} + (2 \beta_{2} + 2) q^{31} + (\beta_{2} - 5) q^{33} + ( - \beta_{3} - 3 \beta_1) q^{35} + (\beta_{2} + 3) q^{37} + (2 \beta_{2} - 2) q^{39} + (2 \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - 2 \beta_{2} - 2) q^{43} + (\beta_{2} - 7) q^{45} + ( - 2 \beta_{3} + \beta_1) q^{47} + ( - \beta_{2} + 2) q^{49} + (4 \beta_{2} - 8) q^{51} + (2 \beta_{3} + 4 \beta_1) q^{53} + ( - 3 \beta_{3} + 3 \beta_1) q^{55} + ( - \beta_{2} + 7) q^{57} + (2 \beta_{2} - 2) q^{59} + ( - 2 \beta_{2} + 6) q^{61} + (2 \beta_{3} + 3 \beta_1) q^{63} + (2 \beta_{3} + 6 \beta_1) q^{65} + (\beta_{3} + 6 \beta_1) q^{67} + (2 \beta_{3} - 6 \beta_1) q^{69} + \beta_1 q^{71} + ( - 3 \beta_{2} - 1) q^{73} + ( - 2 \beta_{3} + 7 \beta_1) q^{75} + ( - 3 \beta_{2} - 3) q^{77} + (2 \beta_{3} - 3 \beta_1) q^{79} + (3 \beta_{2} - 2) q^{81} + 12 q^{83} + 12 \beta_1 q^{85} + ( - 2 \beta_{2} + 6) q^{87} + (6 \beta_{3} + 2 \beta_1) q^{89} + (2 \beta_{2} + 10) q^{91} + (2 \beta_{3} - 2 \beta_1) q^{93} + (5 \beta_{3} - 3 \beta_1) q^{95} - 4 \beta_{3} q^{97} + (4 \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + (-b2 + 1) * q^5 + b3 * q^7 + b2 * q^9 + (b3 + 2b1) q^11 - 2b3 q^13 + (-b3 + 3b1) q^15 + (-2b3 + 2b1) * q^17 + (-2b3 - 3b1) * q^19 + (-b2 + 1) * q^21 + (2b2 - 2) q^23 + (-2b2 + 3) q^25 + (b3 + b1) * q^27 - 2b1 q^29 + (2b2 + 2) q^31 + (b2 - 5) * q^33 + (-b3 - 3b1) q^35 + (b2 + 3) * q^37 + (2b2 - 2) q^39 + (2b2 - 2b1 + 1) * q^41 + (-2b2 - 2) q^43 + (b2 - 7) * q^45 + (-2b3 + b1) q^47 + (-b2 + 2) * q^49 + (4b2 - 8) q^51 + (2b3 + 4b1) * q^53 + (-3b3 + 3b1) * q^55 + (-b2 + 7) * q^57 + (2b2 - 2) q^59 + (-2b2 + 6) q^61 + (2b3 + 3b1) * q^63 + (2b3 + 6b1) * q^65 + (b3 + 6b1) q^67 + (2b3 - 6b1) * q^69 + b1 * q^71 + (-3b2 - 1) q^73 + (-2b3 + 7b1) * q^75 + (-3b2 - 3) q^77 + (2b3 - 3b1) * q^79 + (3b2 - 2) q^81 + 12 * q^83 + 12b1 q^85 + (-2b2 + 6) q^87 + (6b3 + 2b1) * q^89 + (2b2 + 10) q^91 + (2b3 - 2b1) * q^93 + (5b3 - 3b1) * q^95 - 4b3 q^97 + (4b3 - b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} + 4 q^{21} - 8 q^{23} + 12 q^{25} + 8 q^{31} - 20 q^{33} + 12 q^{37} - 8 q^{39} + 4 q^{41} - 8 q^{43} - 28 q^{45} + 8 q^{49} - 32 q^{51} + 28 q^{57} - 8 q^{59} + 24 q^{61} - 4 q^{73} - 12 q^{77} - 8 q^{81} + 48 q^{83} + 24 q^{87} + 40 q^{91}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^21 - 8 * q^23 + 12 * q^25 + 8 * q^31 - 20 * q^33 + 12 * q^37 - 8 * q^39 + 4 * q^41 - 8 * q^43 - 28 * q^45 + 8 * q^49 - 32 * q^51 + 28 * q^57 - 8 * q^59 + 24 * q^61 - 4 * q^73 - 12 * q^77 - 8 * q^81 + 48 * q^83 + 24 * q^87 + 40 * q^91

Display 100 coefficients 10 coefficients

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

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ \nu^{2} + 3 $ v^2 + 3
$\beta_{3}$	$=$	$ \nu^{3} + 5\nu $ v^3 + 5*v

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{2} - 3 $ b2 - 3
$\nu^{3}$	$=$	$ \beta_{3} - 5\beta_1 $ b3 - 5*b1

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

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

	−	2.37608i
	−	0.595188i
		0.595188i
		2.37608i

−

2.37608i

3.64575

1.53436i

−2.64575

2049.2

−

0.595188i

−1.64575

−

2.76510i

2.64575

2049.3

0.595188i

−1.64575

2.76510i

2.64575

2049.4

2.37608i

3.64575

−

1.53436i

−2.64575

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.l		4
4.b	odd	2	1		2624.2.d.m		4
8.b	even	2	1		656.2.d.d		4
8.d	odd	2	1		164.2.b.a	✓	4
24.f	even	2	1		1476.2.f.d		4
40.e	odd	2	1		4100.2.b.e		4
40.k	even	4	2		4100.2.g.c		8
41.b	even	2	1	inner	2624.2.d.l		4
164.d	odd	2	1		2624.2.d.m		4
328.c	odd	2	1		164.2.b.a	✓	4
328.g	even	2	1		656.2.d.d		4
328.k	odd	4	2		6724.2.a.d		4
984.p	even	2	1		1476.2.f.d		4
1640.j	odd	2	1		4100.2.b.e		4
1640.u	even	4	2		4100.2.g.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
164.2.b.a	✓	4	8.d	odd	2	1
164.2.b.a	✓	4	328.c	odd	2	1
656.2.d.d		4	8.b	even	2	1
656.2.d.d		4	328.g	even	2	1
1476.2.f.d		4	24.f	even	2	1
1476.2.f.d		4	984.p	even	2	1
2624.2.d.l		4	1.a	even	1	1	trivial
2624.2.d.l		4	41.b	even	2	1	inner
2624.2.d.m		4	4.b	odd	2	1
2624.2.d.m		4	164.d	odd	2	1
4100.2.b.e		4	40.e	odd	2	1
4100.2.b.e		4	1640.j	odd	2	1
4100.2.g.c		8	40.k	even	4	2
4100.2.g.c		8	1640.u	even	4	2
6724.2.a.d		4	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}^{4} + 6T_{3}^{2} + 2 $

$ T_{7}^{4} + 10T_{7}^{2} + 18 $

$ T_{23}^{2} + 4T_{23} - 24 $

$ T_{31}^{2} - 4T_{31} - 24 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 6T^{2} + 2 $ T^4 + 6*T^2 + 2
$5$	$ (T^{2} - 2 T - 6)^{2} $ (T^2 - 2*T - 6)^2
$7$	$ T^{4} + 10T^{2} + 18 $ T^4 + 10*T^2 + 18
$11$	$ T^{4} + 26T^{2} + 162 $ T^4 + 26*T^2 + 162
$13$	$ T^{4} + 40T^{2} + 288 $ T^4 + 40*T^2 + 288
$17$	$ T^{4} + 80T^{2} + 1152 $ T^4 + 80*T^2 + 1152
$19$	$ T^{4} + 70T^{2} + 882 $ T^4 + 70*T^2 + 882
$23$	$ (T^{2} + 4 T - 24)^{2} $ (T^2 + 4*T - 24)^2
$29$	$ T^{4} + 24T^{2} + 32 $ T^4 + 24*T^2 + 32
$31$	$ (T^{2} - 4 T - 24)^{2} $ (T^2 - 4*T - 24)^2
$37$	$ (T^{2} - 6 T + 2)^{2} $ (T^2 - 6*T + 2)^2
$41$	$ T^{4} - 4 T^{3} + \cdots + 1681 $ T^4 - 4T^3 - 26T^2 - 164*T + 1681
$43$	$ (T^{2} + 4 T - 24)^{2} $ (T^2 + 4*T - 24)^2
$47$	$ T^{4} + 54T^{2} + 722 $ T^4 + 54*T^2 + 722
$53$	$ T^{4} + 104T^{2} + 2592 $ T^4 + 104*T^2 + 2592
$59$	$ (T^{2} + 4 T - 24)^{2} $ (T^2 + 4*T - 24)^2
$61$	$ (T^{2} - 12 T + 8)^{2} $ (T^2 - 12*T + 8)^2
$67$	$ T^{4} + 202T^{2} + 6498 $ T^4 + 202*T^2 + 6498
$71$	$ T^{4} + 6T^{2} + 2 $ T^4 + 6*T^2 + 2
$73$	$ (T^{2} + 2 T - 62)^{2} $ (T^2 + 2*T - 62)^2
$79$	$ T^{4} + 118T^{2} + 1458 $ T^4 + 118*T^2 + 1458
$83$	$ (T - 12)^{4} $ (T - 12)^4
$89$	$ T^{4} + 336T^{2} + 6272 $ T^4 + 336*T^2 + 6272
$97$	$ T^{4} + 160T^{2} + 4608 $ T^4 + 160*T^2 + 4608
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 $\F_{q}$

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

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) q + b1 * q^3 + (-b2 + 1) * q^5 + b3 * q^7 + b2 * q^9	\( q + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{5} + \beta_{3} q^{7} + \beta_{2} q^{9} + (\beta_{3} + 2 \beta_1) q^{11} - 2 \beta_{3} q^{13} + ( - \beta_{3} + 3 \beta_1) q^{15} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{3} - 3 \beta_1) q^{19} + ( - \beta_{2} + 1) q^{21} + (2 \beta_{2} - 2) q^{23} + ( - 2 \beta_{2} + 3) q^{25} + (\beta_{3} + \beta_1) q^{27} - 2 \beta_1 q^{29} + (2 \beta_{2} + 2) q^{31} + (\beta_{2} - 5) q^{33} + ( - \beta_{3} - 3 \beta_1) q^{35} + (\beta_{2} + 3) q^{37} + (2 \beta_{2} - 2) q^{39} + (2 \beta_{2} - 2 \beta_1 + 1) q^{41} + ( - 2 \beta_{2} - 2) q^{43} + (\beta_{2} - 7) q^{45} + ( - 2 \beta_{3} + \beta_1) q^{47} + ( - \beta_{2} + 2) q^{49} + (4 \beta_{2} - 8) q^{51} + (2 \beta_{3} + 4 \beta_1) q^{53} + ( - 3 \beta_{3} + 3 \beta_1) q^{55} + ( - \beta_{2} + 7) q^{57} + (2 \beta_{2} - 2) q^{59} + ( - 2 \beta_{2} + 6) q^{61} + (2 \beta_{3} + 3 \beta_1) q^{63} + (2 \beta_{3} + 6 \beta_1) q^{65} + (\beta_{3} + 6 \beta_1) q^{67} + (2 \beta_{3} - 6 \beta_1) q^{69} + \beta_1 q^{71} + ( - 3 \beta_{2} - 1) q^{73} + ( - 2 \beta_{3} + 7 \beta_1) q^{75} + ( - 3 \beta_{2} - 3) q^{77} + (2 \beta_{3} - 3 \beta_1) q^{79} + (3 \beta_{2} - 2) q^{81} + 12 q^{83} + 12 \beta_1 q^{85} + ( - 2 \beta_{2} + 6) q^{87} + (6 \beta_{3} + 2 \beta_1) q^{89} + (2 \beta_{2} + 10) q^{91} + (2 \beta_{3} - 2 \beta_1) q^{93} + (5 \beta_{3} - 3 \beta_1) q^{95} - 4 \beta_{3} q^{97} + (4 \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + (-b2 + 1) * q^5 + b3 * q^7 + b2 * q^9 + (b3 + 2b1) q^11 - 2b3 q^13 + (-b3 + 3b1) q^15 + (-2b3 + 2b1) * q^17 + (-2b3 - 3b1) * q^19 + (-b2 + 1) * q^21 + (2b2 - 2) q^23 + (-2b2 + 3) q^25 + (b3 + b1) * q^27 - 2b1 q^29 + (2b2 + 2) q^31 + (b2 - 5) * q^33 + (-b3 - 3b1) q^35 + (b2 + 3) * q^37 + (2b2 - 2) q^39 + (2b2 - 2b1 + 1) * q^41 + (-2b2 - 2) q^43 + (b2 - 7) * q^45 + (-2b3 + b1) q^47 + (-b2 + 2) * q^49 + (4b2 - 8) q^51 + (2b3 + 4b1) * q^53 + (-3b3 + 3b1) * q^55 + (-b2 + 7) * q^57 + (2b2 - 2) q^59 + (-2b2 + 6) q^61 + (2b3 + 3b1) * q^63 + (2b3 + 6b1) * q^65 + (b3 + 6b1) q^67 + (2b3 - 6b1) * q^69 + b1 * q^71 + (-3b2 - 1) q^73 + (-2b3 + 7b1) * q^75 + (-3b2 - 3) q^77 + (2b3 - 3b1) * q^79 + (3b2 - 2) q^81 + 12 * q^83 + 12b1 q^85 + (-2b2 + 6) q^87 + (6b3 + 2b1) * q^89 + (2b2 + 10) q^91 + (2b3 - 2b1) * q^93 + (5b3 - 3b1) * q^95 - 4b3 q^97 + (4b3 - b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} + 4 q^{21} - 8 q^{23} + 12 q^{25} + 8 q^{31} - 20 q^{33} + 12 q^{37} - 8 q^{39} + 4 q^{41} - 8 q^{43} - 28 q^{45} + 8 q^{49} - 32 q^{51} + 28 q^{57} - 8 q^{59} + 24 q^{61} - 4 q^{73} - 12 q^{77} - 8 q^{81} + 48 q^{83} + 24 q^{87} + 40 q^{91}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^21 - 8 * q^23 + 12 * q^25 + 8 * q^31 - 20 * q^33 + 12 * q^37 - 8 * q^39 + 4 * q^41 - 8 * q^43 - 28 * q^45 + 8 * q^49 - 32 * q^51 + 28 * q^57 - 8 * q^59 + 24 * q^61 - 4 * q^73 - 12 * q^77 - 8 * q^81 + 48 * q^83 + 24 * q^87 + 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more