LMFDB - Newform orbit 162.4.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,4,Mod(1,162)] 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("162.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	162.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,0,4,21] 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:	$9.55830942093$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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)

$f(q)$

$=$

$ q - 2 q^{2} + 4 q^{4} + 21 q^{5} + 8 q^{7} - 8 q^{8} - 42 q^{10} + 36 q^{11} - 49 q^{13} - 16 q^{14} + 16 q^{16} + 21 q^{17} - 112 q^{19} + 84 q^{20} - 72 q^{22} + 180 q^{23} + 316 q^{25} + 98 q^{26} + 32 q^{28}+ \cdots + 558 q^{98}+O(q^{100}) $

q - 2 * q^2 + 4 * q^4 + 21 * q^5 + 8 * q^7 - 8 * q^8 - 42 * q^10 + 36 * q^11 - 49 * q^13 - 16 * q^14 + 16 * q^16 + 21 * q^17 - 112 * q^19 + 84 * q^20 - 72 * q^22 + 180 * q^23 + 316 * q^25 + 98 * q^26 + 32 * q^28 - 135 * q^29 + 308 * q^31 - 32 * q^32 - 42 * q^34 + 168 * q^35 - q^37 + 224 * q^38 - 168 * q^40 - 42 * q^41 + 20 * q^43 + 144 * q^44 - 360 * q^46 + 84 * q^47 - 279 * q^49 - 632 * q^50 - 196 * q^52 - 174 * q^53 + 756 * q^55 - 64 * q^56 + 270 * q^58 + 504 * q^59 - 385 * q^61 - 616 * q^62 + 64 * q^64 - 1029 * q^65 + 272 * q^67 + 84 * q^68 - 336 * q^70 - 888 * q^71 + 371 * q^73 + 2 * q^74 - 448 * q^76 + 288 * q^77 - 652 * q^79 + 336 * q^80 + 84 * q^82 + 84 * q^83 + 441 * q^85 - 40 * q^86 - 288 * q^88 + 21 * q^89 - 392 * q^91 + 720 * q^92 - 168 * q^94 - 2352 * q^95 - 1246 * q^97 + 558 * q^98

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

−2.00000

4.00000

21.0000

8.00000

−8.00000

−42.0000

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	162.4.a.b	✓	1
3.b	odd	2	1		162.4.a.c	yes	1
4.b	odd	2	1		1296.4.a.h		1
9.c	even	3	2		162.4.c.e		2
9.d	odd	6	2		162.4.c.d		2
12.b	even	2	1		1296.4.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
162.4.a.b	✓	1	1.a	even	1	1	trivial
162.4.a.c	yes	1	3.b	odd	2	1
162.4.c.d		2	9.d	odd	6	2
162.4.c.e		2	9.c	even	3	2
1296.4.a.a		1	12.b	even	2	1
1296.4.a.h		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 21 $ T5 - 21 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(162))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T - 21 $ T - 21
$7$	$ T - 8 $ T - 8
$11$	$ T - 36 $ T - 36
$13$	$ T + 49 $ T + 49
$17$	$ T - 21 $ T - 21
$19$	$ T + 112 $ T + 112
$23$	$ T - 180 $ T - 180
$29$	$ T + 135 $ T + 135
$31$	$ T - 308 $ T - 308
$37$	$ T + 1 $ T + 1
$41$	$ T + 42 $ T + 42
$43$	$ T - 20 $ T - 20
$47$	$ T - 84 $ T - 84
$53$	$ T + 174 $ T + 174
$59$	$ T - 504 $ T - 504
$61$	$ T + 385 $ T + 385
$67$	$ T - 272 $ T - 272
$71$	$ T + 888 $ T + 888
$73$	$ T - 371 $ T - 371
$79$	$ T + 652 $ T + 652
$83$	$ T - 84 $ T - 84
$89$	$ T - 21 $ T - 21
$97$	$ T + 1246 $ T + 1246
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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more