LMFDB - Newform orbit 3600.2.a.bs

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(1,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.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:	$28.7461447277$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of $\beta = 2\sqrt{5}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q+O(q^{10}) $ q	$ q - \beta q^{17} - 4 q^{19} + 2 \beta q^{23} - 8 q^{31} - 2 \beta q^{47} - 7 q^{49} + \beta q^{53} + 2 q^{61} - 16 q^{79} - 4 \beta q^{83} +O(q^{100}) $ q - b * q^17 - 4 * q^19 + 2b q^23 - 8 * q^31 - 2b q^47 - 7 * q^49 + b * q^53 + 2 * q^61 - 16 * q^79 - 4b q^83
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 8 q^{19} - 16 q^{31} - 14 q^{49} + 4 q^{61} - 32 q^{79}+O(q^{100}) $ 2 * q - 8 * q^19 - 16 * q^31 - 14 * q^49 + 4 * q^61 - 32 * q^79

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

1.61803
−0.618034

1.2

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$5$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
3.b	odd	2	1	inner
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3600.2.a.bs		2
3.b	odd	2	1	inner	3600.2.a.bs		2
4.b	odd	2	1		225.2.a.f		2
5.b	even	2	1	inner	3600.2.a.bs		2
5.c	odd	4	2		720.2.f.d		2
12.b	even	2	1		225.2.a.f		2
15.d	odd	2	1	CM	3600.2.a.bs		2
15.e	even	4	2		720.2.f.d		2
20.d	odd	2	1		225.2.a.f		2
20.e	even	4	2		45.2.b.a	✓	2
40.i	odd	4	2		2880.2.f.j		2
40.k	even	4	2		2880.2.f.k		2
60.h	even	2	1		225.2.a.f		2
60.l	odd	4	2		45.2.b.a	✓	2
120.q	odd	4	2		2880.2.f.k		2
120.w	even	4	2		2880.2.f.j		2
140.j	odd	4	2		2205.2.d.a		2
180.v	odd	12	4		405.2.j.c		4
180.x	even	12	4		405.2.j.c		4
420.w	even	4	2		2205.2.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
45.2.b.a	✓	2	20.e	even	4	2
45.2.b.a	✓	2	60.l	odd	4	2
225.2.a.f		2	4.b	odd	2	1
225.2.a.f		2	12.b	even	2	1
225.2.a.f		2	20.d	odd	2	1
225.2.a.f		2	60.h	even	2	1
405.2.j.c		4	180.v	odd	12	4
405.2.j.c		4	180.x	even	12	4
720.2.f.d		2	5.c	odd	4	2
720.2.f.d		2	15.e	even	4	2
2205.2.d.a		2	140.j	odd	4	2
2205.2.d.a		2	420.w	even	4	2
2880.2.f.j		2	40.i	odd	4	2
2880.2.f.j		2	120.w	even	4	2
2880.2.f.k		2	40.k	even	4	2
2880.2.f.k		2	120.q	odd	4	2
3600.2.a.bs		2	1.a	even	1	1	trivial
3600.2.a.bs		2	3.b	odd	2	1	inner
3600.2.a.bs		2	5.b	even	2	1	inner
3600.2.a.bs		2	15.d	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(3600))$:

$ T_{7} $

$ T_{11} $

$ T_{13} $

$ T_{17}^{2} - 20 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} - 20 $ T^2 - 20
$19$	$ (T + 4)^{2} $ (T + 4)^2
$23$	$ T^{2} - 80 $ T^2 - 80
$29$	$ T^{2} $ T^2
$31$	$ (T + 8)^{2} $ (T + 8)^2
$37$	$ T^{2} $ T^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} - 80 $ T^2 - 80
$53$	$ T^{2} - 20 $ T^2 - 20
$59$	$ T^{2} $ T^2
$61$	$ (T - 2)^{2} $ (T - 2)^2
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} $ T^2
$79$	$ (T + 16)^{2} $ (T + 16)^2
$83$	$ T^{2} - 320 $ T^2 - 320
$89$	$ T^{2} $ T^2
$97$	$ T^{2} $ T^2
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Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3600.a (trivial)

\(f(q)\)	\(=\)	\( q+O(q^{10}) \) q	\( q - \beta q^{17} - 4 q^{19} + 2 \beta q^{23} - 8 q^{31} - 2 \beta q^{47} - 7 q^{49} + \beta q^{53} + 2 q^{61} - 16 q^{79} - 4 \beta q^{83} +O(q^{100}) \) q - b * q^17 - 4 * q^19 + 2b q^23 - 8 * q^31 - 2b q^47 - 7 * q^49 + b * q^53 + 2 * q^61 - 16 * q^79 - 4b q^83
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 8 q^{19} - 16 q^{31} - 14 q^{49} + 4 q^{61} - 32 q^{79}+O(q^{100}) \) 2 * q - 8 * q^19 - 16 * q^31 - 14 * q^49 + 4 * q^61 - 32 * q^79

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