LMFDB - Newform orbit 1620.1.f.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,1,Mod(1459,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1620.1459");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1620.f (of order $2$, degree $1$, minimal)

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:	$0.808485320465$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.1620.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.31492800.1
Stark unit:	Root of $x^{6} - 79926x^{5} + 24440847x^{4} - 6339443636x^{3} + 24440847x^{2} - 79926x + 1$

$q$-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}+O(q^{10}) $

$ q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + q^{14} + q^{16} - q^{20} - q^{23} + q^{25} + q^{28} + q^{29} + q^{32} - q^{35} - q^{40} + q^{41} - 2 q^{43} - q^{46} - q^{47} + q^{50} + q^{56} + q^{58} - q^{61} + q^{64} + q^{67} - q^{70} - q^{80} + q^{82} - q^{83} - 2 q^{86} + q^{89} - q^{92} - q^{94}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$0$

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} $

1459.1

1.00000

−1.00000

1.00000

−1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1620.1.f.c		1
3.b	odd	2	1		1620.1.f.b		1
4.b	odd	2	1		1620.1.f.a		1
5.b	even	2	1		1620.1.f.a		1
9.c	even	3	2		540.1.p.a		2
9.d	odd	6	2		180.1.p.b	yes	2
12.b	even	2	1		1620.1.f.d		1
15.d	odd	2	1		1620.1.f.d		1
20.d	odd	2	1	CM	1620.1.f.c		1
36.f	odd	6	2		540.1.p.b		2
36.h	even	6	2		180.1.p.a	✓	2
45.h	odd	6	2		180.1.p.a	✓	2
45.j	even	6	2		540.1.p.b		2
45.k	odd	12	4		2700.1.t.a		4
45.l	even	12	4		900.1.t.a		4
60.h	even	2	1		1620.1.f.b		1
72.j	odd	6	2		2880.1.bu.a		2
72.l	even	6	2		2880.1.bu.b		2
180.n	even	6	2		180.1.p.b	yes	2
180.p	odd	6	2		540.1.p.a		2
180.v	odd	12	4		900.1.t.a		4
180.x	even	12	4		2700.1.t.a		4
360.bd	even	6	2		2880.1.bu.a		2
360.bh	odd	6	2		2880.1.bu.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.1.p.a	✓	2	36.h	even	6	2
180.1.p.a	✓	2	45.h	odd	6	2
180.1.p.b	yes	2	9.d	odd	6	2
180.1.p.b	yes	2	180.n	even	6	2
540.1.p.a		2	9.c	even	3	2
540.1.p.a		2	180.p	odd	6	2
540.1.p.b		2	36.f	odd	6	2
540.1.p.b		2	45.j	even	6	2
900.1.t.a		4	45.l	even	12	4
900.1.t.a		4	180.v	odd	12	4
1620.1.f.a		1	4.b	odd	2	1
1620.1.f.a		1	5.b	even	2	1
1620.1.f.b		1	3.b	odd	2	1
1620.1.f.b		1	60.h	even	2	1
1620.1.f.c		1	1.a	even	1	1	trivial
1620.1.f.c		1	20.d	odd	2	1	CM
1620.1.f.d		1	12.b	even	2	1
1620.1.f.d		1	15.d	odd	2	1
2700.1.t.a		4	45.k	odd	12	4
2700.1.t.a		4	180.x	even	12	4
2880.1.bu.a		2	72.j	odd	6	2
2880.1.bu.a		2	360.bd	even	6	2
2880.1.bu.b		2	72.l	even	6	2
2880.1.bu.b		2	360.bh	odd	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1620, [\chi])$:

$ T_{7} - 1 $

$ T_{23} + 1 $

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 $ T
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T + 1 $ T + 1
$29$	$ T - 1 $ T - 1
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T - 1 $ T - 1
$43$	$ T + 2 $ T + 2
$47$	$ T + 1 $ T + 1
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T - 1 $ T - 1
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T + 1 $ T + 1
$89$	$ T - 1 $ T - 1
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1620.f (of order \(2\), degree \(1\), minimal)

\(n\)	\(811\)	\(1297\)	\(1541\)
\(\chi(n)\)	\(1\)	\(1\)	\(0\)

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