LMFDB - Newform orbit 3249.2.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,-1,0,0,1,-3,0,0,2,0,-5,1,0,-1,4,0,0,0,0,2,4,0,-5,-5,0, -1,-8,0,3,5,0,4,0,0,-3,0,0,0,-12,0,-1,-2,0,4,6,0,-6,-5,0,5,4,0,0,-3,0, -8,10,0,-13,3,0,7,0,0,-11,-4,0,0,6,0,-11,-3,0,0,2,0,-1,0,0,-12,0,0,0,-1, 0,-6,-6,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
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} + q^{7} - 3 q^{8} + 2 q^{11} - 5 q^{13} + q^{14} - q^{16} + 4 q^{17} + 2 q^{22} + 4 q^{23} - 5 q^{25} - 5 q^{26} - q^{28} - 8 q^{29} + 3 q^{31} + 5 q^{32} + 4 q^{34} - 3 q^{37}+ \cdots - 6 q^{98}+O(q^{100}) $

q + q^2 - q^4 + q^7 - 3 * q^8 + 2 * q^11 - 5 * q^13 + q^14 - q^16 + 4 * q^17 + 2 * q^22 + 4 * q^23 - 5 * q^25 - 5 * q^26 - q^28 - 8 * q^29 + 3 * q^31 + 5 * q^32 + 4 * q^34 - 3 * q^37 - 12 * q^41 - q^43 - 2 * q^44 + 4 * q^46 + 6 * q^47 - 6 * q^49 - 5 * q^50 + 5 * q^52 + 4 * q^53 - 3 * q^56 - 8 * q^58 + 10 * q^59 - 13 * q^61 + 3 * q^62 + 7 * q^64 - 11 * q^67 - 4 * q^68 + 6 * q^71 - 11 * q^73 - 3 * q^74 + 2 * q^77 - q^79 - 12 * q^82 - q^86 - 6 * q^88 - 6 * q^89 - 5 * q^91 - 4 * q^92 + 6 * q^94 - 2 * 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

1.00000

−3.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$19$	$ -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	3249.2.a.f		1
3.b	odd	2	1		1083.2.a.b		1
19.b	odd	2	1		3249.2.a.c		1
19.d	odd	6	2		171.2.f.a		2
57.d	even	2	1		1083.2.a.c		1
57.f	even	6	2		57.2.e.a	✓	2
76.f	even	6	2		2736.2.s.j		2
228.n	odd	6	2		912.2.q.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
57.2.e.a	✓	2	57.f	even	6	2
171.2.f.a		2	19.d	odd	6	2
912.2.q.a		2	228.n	odd	6	2
1083.2.a.b		1	3.b	odd	2	1
1083.2.a.c		1	57.d	even	2	1
2736.2.s.j		2	76.f	even	6	2
3249.2.a.c		1	19.b	odd	2	1
3249.2.a.f		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3249))$:

$ T_{2} - 1 $

T2 - 1

$ T_{5} $

T5

$ T_{13} + 5 $

T13 + 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 1 $ T - 1
$11$	$ T - 2 $ T - 2
$13$	$ T + 5 $ T + 5
$17$	$ T - 4 $ T - 4
$19$	$ T $ T
$23$	$ T - 4 $ T - 4
$29$	$ T + 8 $ T + 8
$31$	$ T - 3 $ T - 3
$37$	$ T + 3 $ T + 3
$41$	$ T + 12 $ T + 12
$43$	$ T + 1 $ T + 1
$47$	$ T - 6 $ T - 6
$53$	$ T - 4 $ T - 4
$59$	$ T - 10 $ T - 10
$61$	$ T + 13 $ T + 13
$67$	$ T + 11 $ T + 11
$71$	$ T - 6 $ T - 6
$73$	$ T + 11 $ T + 11
$79$	$ T + 1 $ T + 1
$83$	$ T $ T
$89$	$ T + 6 $ T + 6
$97$	$ T + 2 $ T + 2
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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.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