LMFDB - Newform orbit 3924.1.g.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3924,1,Mod(2179,3924)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3924, 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("3924.2179");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3924 = 2^{2} \cdot 3^{2} \cdot 109 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3924.g (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:	$1.95833110957$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{3}, \sqrt{-109})$
Artin image:	$D_4$
Artin field:	Galois closure of 4.2.47088.1

$q$-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^{8}+O(q^{10}) $

$ q - q^{2} + q^{4} - q^{8} - 2 q^{11} + q^{16} + 2 q^{22} + 2 q^{23} - q^{25} - q^{32} - 2 q^{44} - 2 q^{46} + 2 q^{47} + q^{49} + q^{50} + 2 q^{59} + 2 q^{61} + q^{64} + 2 q^{73} + 2 q^{88} + 2 q^{92} - 2 q^{94} - 2 q^{97} - q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3924\mathbb{Z}\right)^\times$.

$n$	$1963$	$3053$	$3385$
$\chi(n)$	$-1$	$1$	$-1$

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} $

2179.1

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
12.b	even	2	1	RM by $\Q(\sqrt{3}) $
327.d	odd	2	1	CM by $\Q(\sqrt{-327}) $
436.c	odd	2	1	CM by $\Q(\sqrt{-109}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3924.1.g.a	✓	1
3.b	odd	2	1		3924.1.g.c	yes	1
4.b	odd	2	1		3924.1.g.c	yes	1
12.b	even	2	1	RM	3924.1.g.a	✓	1
109.b	even	2	1		3924.1.g.c	yes	1
327.d	odd	2	1	CM	3924.1.g.a	✓	1
436.c	odd	2	1	CM	3924.1.g.a	✓	1
1308.b	even	2	1		3924.1.g.c	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3924.1.g.a	✓	1	1.a	even	1	1	trivial
3924.1.g.a	✓	1	12.b	even	2	1	RM
3924.1.g.a	✓	1	327.d	odd	2	1	CM
3924.1.g.a	✓	1	436.c	odd	2	1	CM
3924.1.g.c	yes	1	3.b	odd	2	1
3924.1.g.c	yes	1	4.b	odd	2	1
3924.1.g.c	yes	1	109.b	even	2	1
3924.1.g.c	yes	1	1308.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3924, [\chi])$:

$ T_{5} $

$ T_{7} $

$ T_{11} + 2 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 1 $ T + 1
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T + 2 $ T + 2
$13$	$ T $ T
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T - 2 $ T - 2
$29$	$ T $ T
$31$	$ T $ T
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T - 2 $ T - 2
$53$	$ T $ T
$59$	$ T - 2 $ T - 2
$61$	$ T - 2 $ T - 2
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T - 2 $ T - 2
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T + 2 $ T + 2
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Level:	\( N \)	\(=\)	\( 3924 = 2^{2} \cdot 3^{2} \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3924.g (of order \(2\), degree \(1\), minimal)

\(n\)	\(1963\)	\(3053\)	\(3385\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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