LMFDB - Newform orbit 48.18.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [48,18,Mod(1,48)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,6561,0,-163554] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$87.9466019254$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3)
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)

$f(q)$

$=$

$ q + 6561 q^{3} - 163554 q^{5} + 20846560 q^{7} + 43046721 q^{9} - 817372356 q^{11} + 299589758 q^{13} - 1073077794 q^{15} - 44775606078 q^{17} - 78748651964 q^{19} + 136774280160 q^{21} + 704672009160 q^{23}+ \cdots - 35\!\cdots\!76 q^{99}+O(q^{100}) $

q + 6561 * q^3 - 163554 * q^5 + 20846560 * q^7 + 43046721 * q^9 - 817372356 * q^11 + 299589758 * q^13 - 1073077794 * q^15 - 44775606078 * q^17 - 78748651964 * q^19 + 136774280160 * q^21 + 704672009160 * q^23 - 736189542209 * q^25 + 282429536481 * q^27 - 163793785242 * q^29 - 1049860831400 * q^31 - 5362780027716 * q^33 - 3409538274240 * q^35 - 19805735857210 * q^37 + 1965608402238 * q^39 + 14660035932090 * q^41 - 116038864682564 * q^43 - 7040463406434 * q^45 + 176606594594112 * q^47 + 201948549846393 * q^49 - 293772751477758 * q^51 + 152863496635230 * q^53 + 133684518313224 * q^55 - 516669905535804 * q^57 + 262797291296124 * q^59 - 1358552281482562 * q^61 + 897376052129760 * q^63 - 48999103279932 * q^65 - 444863620615292 * q^67 + 4623353052098760 * q^69 + 4003270764790968 * q^71 + 924832535317130 * q^73 - 4830139586433249 * q^75 - 17039401861695360 * q^77 - 14747307742797080 * q^79 + 1853020188851841 * q^81 - 26422963268810172 * q^83 + 7323229476481212 * q^85 - 1074651024972762 * q^87 - 38883748080645126 * q^89 + 6245415865532480 * q^91 - 6888136914815400 * q^93 + 12879657023320056 * q^95 - 25374394856250238 * q^97 - 35185199761844676 * q^99

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

6561.00

−163554.

2.08466e7

4.30467e7

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -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	48.18.a.e		1
4.b	odd	2	1		3.18.a.a	✓	1
12.b	even	2	1		9.18.a.a		1
20.d	odd	2	1		75.18.a.a		1
20.e	even	4	2		75.18.b.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3.18.a.a	✓	1	4.b	odd	2	1
9.18.a.a		1	12.b	even	2	1
48.18.a.e		1	1.a	even	1	1	trivial
75.18.a.a		1	20.d	odd	2	1
75.18.b.a		2	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 163554 $ T5 + 163554 acting on $S_{18}^{\mathrm{new}}(\Gamma_0(48))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 6561 $ T - 6561
$5$	$ T + 163554 $ T + 163554
$7$	$ T - 20846560 $ T - 20846560
$11$	$ T + 817372356 $ T + 817372356
$13$	$ T - 299589758 $ T - 299589758
$17$	$ T + 44775606078 $ T + 44775606078
$19$	$ T + 78748651964 $ T + 78748651964
$23$	$ T - 704672009160 $ T - 704672009160
$29$	$ T + 163793785242 $ T + 163793785242
$31$	$ T + 1049860831400 $ T + 1049860831400
$37$	$ T + 19805735857210 $ T + 19805735857210
$41$	$ T - 14660035932090 $ T - 14660035932090
$43$	$ T + 116038864682564 $ T + 116038864682564
$47$	$ T - 176606594594112 $ T - 176606594594112
$53$	$ T - 152863496635230 $ T - 152863496635230
$59$	$ T - 262797291296124 $ T - 262797291296124
$61$	$ T + 1358552281482562 $ T + 1358552281482562
$67$	$ T + 444863620615292 $ T + 444863620615292
$71$	$ T - 4003270764790968 $ T - 4003270764790968
$73$	$ T - 924832535317130 $ T - 924832535317130
$79$	$ T + 14\!\cdots\!80 $ T + 14747307742797080
$83$	$ T + 26\!\cdots\!72 $ T + 26422963268810172
$89$	$ T + 38\!\cdots\!26 $ T + 38883748080645126
$97$	$ T + 25\!\cdots\!38 $ T + 25374394856250238
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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more