LMFDB - Newform orbit 4680.2.l.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4680,2,Mod(2809,4680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4680.2809"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4680.l (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 4x^{3} + 25x^{2} - 20x + 8$ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 1560)
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{4} q^{5} + ( - \beta_{5} - \beta_{4} + 2) q^{11} - \beta_{3} q^{13} + ( - \beta_{5} + \beta_{4} + \cdots - \beta_1) q^{17} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{19} + (\beta_{5} - \beta_{4} - 4 \beta_{3} + \cdots + \beta_1) q^{23}+ \cdots - 6 \beta_{3} q^{97}+O(q^{100})$ q + b4 * q^5 + (-b5 - b4 + 2) * q^11 - b3 * q^13 + (-b5 + b4 + 2b3 - b2 - b1) q^17 + (-b5 - b4 + b2 - b1) * q^19 + (b5 - b4 - 4b3 + b2 + b1) q^23 + (b5 - b3 + 2b2 + b1 - 2) q^25 + (-2b2 + 2b1 - 2) * q^29 + (-2b5 - 2b4 + 4) * q^31 + (2b3 + 2b2 + 2b1) q^37 + (-2b5 - 2b4 - b2 + b1 + 4) * q^41 + (b5 - b4 - 2b3 + b2 + b1) q^43 + (-b5 + b4 + 6b3 - 2b2 - 2b1) q^47 + 7 * q^49 + (-2b5 + 2b4 - 2b3) q^53 + (-b5 + 2b4 + b3 - 2b2 - b1 - 3) * q^55 + (-3b5 - 3b4 - 2) * q^59 + (b5 + b4 + b2 - b1) * q^61 + b1 * q^65 + (-2b5 + 2b4 + 2b2 + 2b1) * q^67 + (-b2 + b1 - 2) * q^71 + (2b3 + 4b2 + 4b1) q^73 + (b5 + b4 - b2 + b1 - 6) * q^79 + (b5 - b4 - 2b3 - 2b2 - 2b1) q^83 + (3b5 + b4 + 2b3 + b2 - b1 - 6) * q^85 + (2b5 + 2b4 + b2 - b1 - 8) * q^89 + (b5 + b4 - 6b3 - 3b2 - b1 - 2) * q^95 - 6b3 q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100})$ 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 4x^{3} + 25x^{2} - 20x + 8$ x^6 - 4*x^3 + 25*x^2 - 20*x + 8 :

$\beta_{1}$	$=$	$( 2\nu^{5} + 25\nu^{4} + 10\nu^{3} - 4\nu^{2} - 121\nu + 323 ) / 121$ (2v^5 + 25v^4 + 10v^3 - 4v^2 - 121*v + 323) / 121
$\beta_{2}$	$=$	$( -7\nu^{5} - 27\nu^{4} - 35\nu^{3} + 14\nu^{2} - 121\nu - 223 ) / 121$ (-7v^5 - 27v^4 - 35v^3 + 14v^2 - 121*v - 223) / 121
$\beta_{3}$	$=$	$( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242$ (-25v^5 - 10v^4 - 4v^3 + 50v^2 - 605*v + 258) / 242
$\beta_{4}$	$=$	$( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1573\nu + 574 ) / 242$ (-65v^5 - 26v^4 + 38v^3 + 372v^2 - 1573*v + 574) / 242
$\beta_{5}$	$=$	$( 75\nu^{5} + 30\nu^{4} + 12\nu^{3} - 392\nu^{2} + 1573\nu - 774 ) / 242$ (75v^5 + 30v^4 + 12v^3 - 392v^2 + 1573*v - 774) / 242

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

$\nu$	$=$	$( -\beta_{5} - \beta_{4} - \beta_{2} - \beta_1 ) / 2$ (-b5 - b4 - b2 - b1) / 2
$\nu^{2}$	$=$	$( -\beta_{5} + \beta_{4} - 6\beta_{3} + \beta_{2} + \beta_1 ) / 2$ (-b5 + b4 - 6*b3 + b2 + b1) / 2
$\nu^{3}$	$=$	$( 5\beta_{5} + 5\beta_{4} + 4\beta_{3} - 5\beta_{2} - 5\beta _1 + 4 ) / 2$ (5b5 + 5b4 + 4b3 - 5b2 - 5*b1 + 4) / 2
$\nu^{4}$	$=$	$( -9\beta_{5} - 9\beta_{4} - 5\beta_{2} + 5\beta _1 - 30 ) / 2$ (-9b5 - 9b4 - 5b2 + 5b1 - 30) / 2
$\nu^{5}$	$=$	$( 25\beta_{5} + 29\beta_{4} - 32\beta_{3} + 29\beta_{2} + 25\beta _1 + 32 ) / 2$ (25b5 + 29b4 - 32b3 + 29b2 + 25*b1 + 32) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$ .

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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}

2809.1

1.32001	−	1.32001i
1.32001	+	1.32001i
0.432320	+	0.432320i
0.432320	−	0.432320i
−1.75233	+	1.75233i
−1.75233	−	1.75233i

−1.32001

−

1.80487i

2809.2

−1.32001

1.80487i

2809.3

−0.432320

−

2.19388i

2809.4

−0.432320

2.19388i

2809.5

1.75233

−

1.38900i

2809.6

1.75233

1.38900i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4680.2.l.e		6
3.b	odd	2	1		1560.2.l.c	✓	6
5.b	even	2	1	inner	4680.2.l.e		6
12.b	even	2	1		3120.2.l.m		6
15.d	odd	2	1		1560.2.l.c	✓	6
15.e	even	4	1		7800.2.a.bj		3
15.e	even	4	1		7800.2.a.bp		3
60.h	even	2	1		3120.2.l.m		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1560.2.l.c	✓	6	3.b	odd	2	1
1560.2.l.c	✓	6	15.d	odd	2	1
3120.2.l.m		6	12.b	even	2	1
3120.2.l.m		6	60.h	even	2	1
4680.2.l.e		6	1.a	even	1	1	trivial
4680.2.l.e		6	5.b	even	2	1	inner
7800.2.a.bj		3	15.e	even	4	1
7800.2.a.bp		3	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4680, [\chi])$ :

T_{7}

T7

T_{11}^{3} - 6T_{11}^{2} + 2T_{11} + 20

T11^3 - 6*T11^2 + 2*T11 + 20

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6}$ T^6
$5$	$T^{6} + 5 T^{4} + \cdots + 125$ T^6 + 5T^4 - 8T^3 + 25*T^2 + 125
$7$	$T^{6}$ T^6
$11$	$(T^{3} - 6 T^{2} + 2 T + 20)^{2}$ (T^3 - 6T^2 + 2T + 20)^2
$13$	$(T^{2} + 1)^{3}$ (T^2 + 1)^3
$17$	$T^{6} + 56 T^{4} + \cdots + 4096$ T^6 + 56T^4 + 912T^2 + 4096
$19$	$(T^{3} - 2 T^{2} - 24 T - 16)^{2}$ (T^3 - 2T^2 - 24T - 16)^2
$23$	$T^{6} + 84 T^{4} + \cdots + 6400$ T^6 + 84T^4 + 1664T^2 + 6400
$29$	$(T^{3} + 10 T^{2} + \cdots - 472)^{2}$ (T^3 + 10T^2 - 44T - 472)^2
$31$	$(T^{3} - 12 T^{2} + \cdots + 160)^{2}$ (T^3 - 12T^2 + 8T + 160)^2
$37$	$T^{6} + 92 T^{4} + \cdots + 18496$ T^6 + 92T^4 + 2416T^2 + 18496
$41$	$(T^{3} - 10 T^{2} + \cdots + 332)^{2}$ (T^3 - 10T^2 - 34T + 332)^2
$43$	$T^{6} + 56 T^{4} + \cdots + 4096$ T^6 + 56T^4 + 912T^2 + 4096
$47$	$T^{6} + 188 T^{4} + \cdots + 65536$ T^6 + 188T^4 + 9348T^2 + 65536
$53$	$T^{6} + 188 T^{4} + \cdots + 222784$ T^6 + 188T^4 + 11376T^2 + 222784
$59$	$(T^{3} + 6 T^{2} - 78 T + 44)^{2}$ (T^3 + 6T^2 - 78T + 44)^2
$61$	$(T^{3} - 2 T^{2} - 32 T + 32)^{2}$ (T^3 - 2T^2 - 32T + 32)^2
$67$	$T^{6} + 272 T^{4} + \cdots + 65536$ T^6 + 272T^4 + 18432T^2 + 65536
$71$	$(T^{3} + 8 T^{2} + 2 T - 64)^{2}$ (T^3 + 8T^2 + 2T - 64)^2
$73$	$T^{6} + 332 T^{4} + \cdots + 678976$ T^6 + 332T^4 + 31792T^2 + 678976
$79$	$(T^{3} + 20 T^{2} + \cdots + 160)^{2}$ (T^3 + 20T^2 + 108T + 160)^2
$83$	$T^{6} + 140 T^{4} + \cdots + 53824$ T^6 + 140T^4 + 5700T^2 + 53824
$89$	$(T^{3} + 22 T^{2} + \cdots - 244)^{2}$ (T^3 + 22T^2 + 94T - 244)^2
$97$	$(T^{2} + 36)^{3}$ (T^2 + 36)^3
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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 2809.6
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more