LMFDB - Newform orbit 60.8.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [60,8,Mod(1,60)] 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("60.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	60.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-27,0,-125] 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:	$18.7431015290$
Analytic rank:	$1$
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 - 27 q^{3} - 125 q^{5} + 1028 q^{7} + 729 q^{9} + 3096 q^{11} - 13030 q^{13} + 3375 q^{15} + 1878 q^{17} - 31180 q^{19} - 27756 q^{21} - 33288 q^{23} + 15625 q^{25} - 19683 q^{27} - 213054 q^{29} - 172696 q^{31}+ \cdots + 2256984 q^{99}+O(q^{100}) $

q - 27 * q^3 - 125 * q^5 + 1028 * q^7 + 729 * q^9 + 3096 * q^11 - 13030 * q^13 + 3375 * q^15 + 1878 * q^17 - 31180 * q^19 - 27756 * q^21 - 33288 * q^23 + 15625 * q^25 - 19683 * q^27 - 213054 * q^29 - 172696 * q^31 - 83592 * q^33 - 128500 * q^35 + 27434 * q^37 + 351810 * q^39 + 532650 * q^41 - 911908 * q^43 - 91125 * q^45 - 732648 * q^47 + 233241 * q^49 - 50706 * q^51 + 409074 * q^53 - 387000 * q^55 + 841860 * q^57 + 1508136 * q^59 - 302578 * q^61 + 749412 * q^63 + 1628750 * q^65 + 1254332 * q^67 + 898776 * q^69 + 4781280 * q^71 - 502414 * q^73 - 421875 * q^75 + 3182688 * q^77 - 1991368 * q^79 + 531441 * q^81 - 8099268 * q^83 - 234750 * q^85 + 5752458 * q^87 + 7487970 * q^89 - 13394840 * q^91 + 4662792 * q^93 + 3897500 * q^95 - 17172574 * q^97 + 2256984 * 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

−27.0000

−125.000

1028.00

729.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ +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	60.8.a.a	✓	1
3.b	odd	2	1		180.8.a.e		1
4.b	odd	2	1		240.8.a.i		1
5.b	even	2	1		300.8.a.e		1
5.c	odd	4	2		300.8.d.d		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
60.8.a.a	✓	1	1.a	even	1	1	trivial
180.8.a.e		1	3.b	odd	2	1
240.8.a.i		1	4.b	odd	2	1
300.8.a.e		1	5.b	even	2	1
300.8.d.d		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} - 1028 $ T7 - 1028 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(60))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 27 $ T + 27
$5$	$ T + 125 $ T + 125
$7$	$ T - 1028 $ T - 1028
$11$	$ T - 3096 $ T - 3096
$13$	$ T + 13030 $ T + 13030
$17$	$ T - 1878 $ T - 1878
$19$	$ T + 31180 $ T + 31180
$23$	$ T + 33288 $ T + 33288
$29$	$ T + 213054 $ T + 213054
$31$	$ T + 172696 $ T + 172696
$37$	$ T - 27434 $ T - 27434
$41$	$ T - 532650 $ T - 532650
$43$	$ T + 911908 $ T + 911908
$47$	$ T + 732648 $ T + 732648
$53$	$ T - 409074 $ T - 409074
$59$	$ T - 1508136 $ T - 1508136
$61$	$ T + 302578 $ T + 302578
$67$	$ T - 1254332 $ T - 1254332
$71$	$ T - 4781280 $ T - 4781280
$73$	$ T + 502414 $ T + 502414
$79$	$ T + 1991368 $ T + 1991368
$83$	$ T + 8099268 $ T + 8099268
$89$	$ T - 7487970 $ T - 7487970
$97$	$ T + 17172574 $ T + 17172574
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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	60.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