LMFDB - Newform orbit 1710.4.a.g

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1710,4,Mod(1,1710)] 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("1710.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1710, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1710.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,4,-5,0,-12,8,0,-10,20] 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:	$100.893266110$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
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} - 5 q^{5} - 12 q^{7} + 8 q^{8} - 10 q^{10} + 20 q^{11} - 4 q^{13} - 24 q^{14} + 16 q^{16} + 34 q^{17} - 19 q^{19} - 20 q^{20} + 40 q^{22} - 40 q^{23} + 25 q^{25} - 8 q^{26} - 48 q^{28}+ \cdots - 398 q^{98}+O(q^{100}) $

q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 12 * q^7 + 8 * q^8 - 10 * q^10 + 20 * q^11 - 4 * q^13 - 24 * q^14 + 16 * q^16 + 34 * q^17 - 19 * q^19 - 20 * q^20 + 40 * q^22 - 40 * q^23 + 25 * q^25 - 8 * q^26 - 48 * q^28 + 150 * q^29 - 200 * q^31 + 32 * q^32 + 68 * q^34 + 60 * q^35 - 156 * q^37 - 38 * q^38 - 40 * q^40 + 218 * q^41 + 248 * q^43 + 80 * q^44 - 80 * q^46 + 180 * q^47 - 199 * q^49 + 50 * q^50 - 16 * q^52 - 72 * q^53 - 100 * q^55 - 96 * q^56 + 300 * q^58 + 48 * q^59 - 134 * q^61 - 400 * q^62 + 64 * q^64 + 20 * q^65 + 334 * q^67 + 136 * q^68 + 120 * q^70 + 520 * q^71 + 438 * q^73 - 312 * q^74 - 76 * q^76 - 240 * q^77 + 980 * q^79 - 80 * q^80 + 436 * q^82 + 156 * q^83 - 170 * q^85 + 496 * q^86 + 160 * q^88 - 670 * q^89 + 48 * q^91 - 160 * q^92 + 360 * q^94 + 95 * q^95 + 1124 * q^97 - 398 * 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

−5.00000

−12.0000

8.00000

−10.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$5$	$ +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	1710.4.a.g		1
3.b	odd	2	1		190.4.a.b	✓	1
12.b	even	2	1		1520.4.a.d		1
15.d	odd	2	1		950.4.a.b		1
15.e	even	4	2		950.4.b.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
190.4.a.b	✓	1	3.b	odd	2	1
950.4.a.b		1	15.d	odd	2	1
950.4.b.b		2	15.e	even	4	2
1520.4.a.d		1	12.b	even	2	1
1710.4.a.g		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 $S_{4}^{\mathrm{new}}(\Gamma_0(1710))$:

$ T_{7} + 12 $

T7 + 12

$ T_{11} - 20 $

T11 - 20

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T $ T
$5$	$ T + 5 $ T + 5
$7$	$ T + 12 $ T + 12
$11$	$ T - 20 $ T - 20
$13$	$ T + 4 $ T + 4
$17$	$ T - 34 $ T - 34
$19$	$ T + 19 $ T + 19
$23$	$ T + 40 $ T + 40
$29$	$ T - 150 $ T - 150
$31$	$ T + 200 $ T + 200
$37$	$ T + 156 $ T + 156
$41$	$ T - 218 $ T - 218
$43$	$ T - 248 $ T - 248
$47$	$ T - 180 $ T - 180
$53$	$ T + 72 $ T + 72
$59$	$ T - 48 $ T - 48
$61$	$ T + 134 $ T + 134
$67$	$ T - 334 $ T - 334
$71$	$ T - 520 $ T - 520
$73$	$ T - 438 $ T - 438
$79$	$ T - 980 $ T - 980
$83$	$ T - 156 $ T - 156
$89$	$ T + 670 $ T + 670
$97$	$ T - 1124 $ T - 1124
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:	\( N \)	\(=\)	\( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1710.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