LMFDB - Newform orbit 2142.2.a.r

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2142,2,Mod(1,2142)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2142, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2142.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2142.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-1,0,1,1,0,-1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.1039561130$
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 + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} - 3 q^{11} - 5 q^{13} + q^{14} + q^{16} - q^{17} - 6 q^{19} - q^{20} - 3 q^{22} + 8 q^{23} - 4 q^{25} - 5 q^{26} + q^{28} - 4 q^{29} - 6 q^{31}+ \cdots + q^{98}+O(q^{100}) $

q + q^2 + q^4 - q^5 + q^7 + q^8 - q^10 - 3 * q^11 - 5 * q^13 + q^14 + q^16 - q^17 - 6 * q^19 - q^20 - 3 * q^22 + 8 * q^23 - 4 * q^25 - 5 * q^26 + q^28 - 4 * q^29 - 6 * q^31 + q^32 - q^34 - q^35 - 5 * q^37 - 6 * q^38 - q^40 + 4 * q^41 + q^43 - 3 * q^44 + 8 * q^46 - 2 * q^47 + q^49 - 4 * q^50 - 5 * q^52 + 3 * q^53 + 3 * q^55 + q^56 - 4 * q^58 - 4 * q^59 - 8 * q^61 - 6 * q^62 + q^64 + 5 * q^65 - q^67 - q^68 - q^70 - 8 * q^71 - 5 * q^73 - 5 * q^74 - 6 * q^76 - 3 * q^77 + 9 * q^79 - q^80 + 4 * q^82 + 9 * q^83 + q^85 + q^86 - 3 * q^88 + 7 * q^89 - 5 * q^91 + 8 * q^92 - 2 * q^94 + 6 * q^95 + 3 * q^97 + 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

−1.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$7$	$ -1 $
$17$	$ +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	2142.2.a.r	yes	1
3.b	odd	2	1		2142.2.a.f	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2142.2.a.f	✓	1	3.b	odd	2	1
2142.2.a.r	yes	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 $S_{2}^{\mathrm{new}}(\Gamma_0(2142))$:

$ T_{5} + 1 $

T5 + 1

$ T_{11} + 3 $

T11 + 3

$ T_{13} + 5 $

T13 + 5

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T + 1 $ T + 1
$7$	$ T - 1 $ T - 1
$11$	$ T + 3 $ T + 3
$13$	$ T + 5 $ T + 5
$17$	$ T + 1 $ T + 1
$19$	$ T + 6 $ T + 6
$23$	$ T - 8 $ T - 8
$29$	$ T + 4 $ T + 4
$31$	$ T + 6 $ T + 6
$37$	$ T + 5 $ T + 5
$41$	$ T - 4 $ T - 4
$43$	$ T - 1 $ T - 1
$47$	$ T + 2 $ T + 2
$53$	$ T - 3 $ T - 3
$59$	$ T + 4 $ T + 4
$61$	$ T + 8 $ T + 8
$67$	$ T + 1 $ T + 1
$71$	$ T + 8 $ T + 8
$73$	$ T + 5 $ T + 5
$79$	$ T - 9 $ T - 9
$83$	$ T - 9 $ T - 9
$89$	$ T - 7 $ T - 7
$97$	$ T - 3 $ T - 3
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Level:	\( N \)	\(=\)	\( 2142 = 2 \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2142.a (trivial)

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