LMFDB - Newform orbit 9126.2.a.r

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9126,2,Mod(1,9126)] 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("9126.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9126 = 2 \cdot 3^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,0,1,3,0,1,-1,0,-3,-3,0,0,-1,0,1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.8714768846$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
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 - q^{2} + q^{4} + 3 q^{5} + q^{7} - q^{8} - 3 q^{10} - 3 q^{11} - q^{14} + q^{16} - 2 q^{19} + 3 q^{20} + 3 q^{22} + 6 q^{23} + 4 q^{25} + q^{28} - 6 q^{29} - 5 q^{31} - q^{32} + 3 q^{35} - 2 q^{37}+ \cdots + 6 q^{98}+O(q^{100}) $

q - q^2 + q^4 + 3 * q^5 + q^7 - q^8 - 3 * q^10 - 3 * q^11 - q^14 + q^16 - 2 * q^19 + 3 * q^20 + 3 * q^22 + 6 * q^23 + 4 * q^25 + q^28 - 6 * q^29 - 5 * q^31 - q^32 + 3 * q^35 - 2 * q^37 + 2 * q^38 - 3 * q^40 - 6 * q^41 - 10 * q^43 - 3 * q^44 - 6 * q^46 + 6 * q^47 - 6 * q^49 - 4 * q^50 - 9 * q^53 - 9 * q^55 - q^56 + 6 * q^58 + 12 * q^59 + 8 * q^61 + 5 * q^62 + q^64 - 14 * q^67 - 3 * q^70 + 7 * q^73 + 2 * q^74 - 2 * q^76 - 3 * q^77 + 8 * q^79 + 3 * q^80 + 6 * q^82 - 3 * q^83 + 10 * q^86 + 3 * q^88 - 18 * q^89 + 6 * q^92 - 6 * q^94 - 6 * q^95 + q^97 + 6 * 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

−1.00000

1.00000

3.00000

1.00000

−1.00000

−3.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$13$	$ +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	9126.2.a.r		1
3.b	odd	2	1		9126.2.a.u		1
13.b	even	2	1		54.2.a.b	yes	1
39.d	odd	2	1		54.2.a.a	✓	1
52.b	odd	2	1		432.2.a.b		1
65.d	even	2	1		1350.2.a.h		1
65.h	odd	4	2		1350.2.c.k		2
91.b	odd	2	1		2646.2.a.bd		1
104.e	even	2	1		1728.2.a.y		1
104.h	odd	2	1		1728.2.a.z		1
117.n	odd	6	2		162.2.c.c		2
117.t	even	6	2		162.2.c.b		2
143.d	odd	2	1		6534.2.a.b		1
156.h	even	2	1		432.2.a.g		1
195.e	odd	2	1		1350.2.a.r		1
195.s	even	4	2		1350.2.c.b		2
273.g	even	2	1		2646.2.a.a		1
312.b	odd	2	1		1728.2.a.c		1
312.h	even	2	1		1728.2.a.d		1
429.e	even	2	1		6534.2.a.bc		1
468.x	even	6	2		1296.2.i.c		2
468.bg	odd	6	2		1296.2.i.o		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.2.a.a	✓	1	39.d	odd	2	1
54.2.a.b	yes	1	13.b	even	2	1
162.2.c.b		2	117.t	even	6	2
162.2.c.c		2	117.n	odd	6	2
432.2.a.b		1	52.b	odd	2	1
432.2.a.g		1	156.h	even	2	1
1296.2.i.c		2	468.x	even	6	2
1296.2.i.o		2	468.bg	odd	6	2
1350.2.a.h		1	65.d	even	2	1
1350.2.a.r		1	195.e	odd	2	1
1350.2.c.b		2	195.s	even	4	2
1350.2.c.k		2	65.h	odd	4	2
1728.2.a.c		1	312.b	odd	2	1
1728.2.a.d		1	312.h	even	2	1
1728.2.a.y		1	104.e	even	2	1
1728.2.a.z		1	104.h	odd	2	1
2646.2.a.a		1	273.g	even	2	1
2646.2.a.bd		1	91.b	odd	2	1
6534.2.a.b		1	143.d	odd	2	1
6534.2.a.bc		1	429.e	even	2	1
9126.2.a.r		1	1.a	even	1	1	trivial
9126.2.a.u		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9126))$:

$ T_{5} - 3 $

T5 - 3

$ T_{7} - 1 $

T7 - 1

$ T_{11} + 3 $

T11 + 3

$ T_{17} $

T17

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T $ T
$5$	$ T - 3 $ T - 3
$7$	$ T - 1 $ T - 1
$11$	$ T + 3 $ T + 3
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T + 2 $ T + 2
$23$	$ T - 6 $ T - 6
$29$	$ T + 6 $ T + 6
$31$	$ T + 5 $ T + 5
$37$	$ T + 2 $ T + 2
$41$	$ T + 6 $ T + 6
$43$	$ T + 10 $ T + 10
$47$	$ T - 6 $ T - 6
$53$	$ T + 9 $ T + 9
$59$	$ T - 12 $ T - 12
$61$	$ T - 8 $ T - 8
$67$	$ T + 14 $ T + 14
$71$	$ T $ T
$73$	$ T - 7 $ T - 7
$79$	$ T - 8 $ T - 8
$83$	$ T + 3 $ T + 3
$89$	$ T + 18 $ T + 18
$97$	$ T - 1 $ T - 1
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:	\( N \)	\(=\)	\( 9126 = 2 \cdot 3^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9126.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