LMFDB - Newform orbit 1764.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(1,1764)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1764.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$104.079369250$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 588)
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 + 4 q^{5}+O(q^{10}) $

$ q + 4 q^{5} + 20 q^{11} + 4 q^{13} + 24 q^{17} - 44 q^{19} - 72 q^{23} - 109 q^{25} + 38 q^{29} - 184 q^{31} - 30 q^{37} - 216 q^{41} - 164 q^{43} + 520 q^{47} + 146 q^{53} + 80 q^{55} + 460 q^{59} - 628 q^{61} + 16 q^{65} + 556 q^{67} - 592 q^{71} - 1024 q^{73} - 104 q^{79} - 324 q^{83} + 96 q^{85} + 896 q^{89} - 176 q^{95} + 920 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

4.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$-1$
$7$	$-1$

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	1764.4.a.i		1
3.b	odd	2	1		588.4.a.e	yes	1
7.b	odd	2	1		1764.4.a.d		1
7.c	even	3	2		1764.4.k.g		2
7.d	odd	6	2		1764.4.k.j		2
12.b	even	2	1		2352.4.a.g		1
21.c	even	2	1		588.4.a.b	✓	1
21.g	even	6	2		588.4.i.g		2
21.h	odd	6	2		588.4.i.b		2
84.h	odd	2	1		2352.4.a.be		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
588.4.a.b	✓	1	21.c	even	2	1
588.4.a.e	yes	1	3.b	odd	2	1
588.4.i.b		2	21.h	odd	6	2
588.4.i.g		2	21.g	even	6	2
1764.4.a.d		1	7.b	odd	2	1
1764.4.a.i		1	1.a	even	1	1	trivial
1764.4.k.g		2	7.c	even	3	2
1764.4.k.j		2	7.d	odd	6	2
2352.4.a.g		1	12.b	even	2	1
2352.4.a.be		1	84.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1764))$:

$ T_{5} - 4 $

$ T_{11} - 20 $

$ T_{13} - 4 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T - 4 $ T - 4
$7$	$ T $ T
$11$	$ T - 20 $ T - 20
$13$	$ T - 4 $ T - 4
$17$	$ T - 24 $ T - 24
$19$	$ T + 44 $ T + 44
$23$	$ T + 72 $ T + 72
$29$	$ T - 38 $ T - 38
$31$	$ T + 184 $ T + 184
$37$	$ T + 30 $ T + 30
$41$	$ T + 216 $ T + 216
$43$	$ T + 164 $ T + 164
$47$	$ T - 520 $ T - 520
$53$	$ T - 146 $ T - 146
$59$	$ T - 460 $ T - 460
$61$	$ T + 628 $ T + 628
$67$	$ T - 556 $ T - 556
$71$	$ T + 592 $ T + 592
$73$	$ T + 1024 $ T + 1024
$79$	$ T + 104 $ T + 104
$83$	$ T + 324 $ T + 324
$89$	$ T - 896 $ T - 896
$97$	$ T - 920 $ T - 920
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

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