LMFDB - Newform orbit 324.1.d.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,1,Mod(163,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("324.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	324.d (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:	$0.161697064093$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.324.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.324.1

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

$ q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - q^{13} + q^{16} - q^{17} - q^{20} - q^{26} - q^{29} + q^{32} - q^{34} - q^{37} - q^{40} + 2 q^{41} + q^{49} - q^{52} + 2 q^{53} - q^{58} - q^{61} + q^{64} + q^{65} - q^{68} - q^{73} - q^{74} - q^{80} + 2 q^{82} + q^{85} - q^{89} + 2 q^{97} + q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$1$

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} $

163.1

1.00000

−1.00000

1.00000

−1.00000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.1.d.b	yes	1
3.b	odd	2	1		324.1.d.a	✓	1
4.b	odd	2	1	CM	324.1.d.b	yes	1
9.c	even	3	2		324.1.f.a		2
9.d	odd	6	2		324.1.f.b		2
12.b	even	2	1		324.1.d.a	✓	1
27.e	even	9	6		2916.1.j.d		6
27.f	odd	18	6		2916.1.j.e		6
36.f	odd	6	2		324.1.f.a		2
36.h	even	6	2		324.1.f.b		2
108.j	odd	18	6		2916.1.j.d		6
108.l	even	18	6		2916.1.j.e		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
324.1.d.a	✓	1	3.b	odd	2	1
324.1.d.a	✓	1	12.b	even	2	1
324.1.d.b	yes	1	1.a	even	1	1	trivial
324.1.d.b	yes	1	4.b	odd	2	1	CM
324.1.f.a		2	9.c	even	3	2
324.1.f.a		2	36.f	odd	6	2
324.1.f.b		2	9.d	odd	6	2
324.1.f.b		2	36.h	even	6	2
2916.1.j.d		6	27.e	even	9	6
2916.1.j.d		6	108.j	odd	18	6
2916.1.j.e		6	27.f	odd	18	6
2916.1.j.e		6	108.l	even	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 1 $ T5 + 1 acting on $S_{1}^{\mathrm{new}}(324, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T + 1 $ T + 1
$17$	$ T + 1 $ T + 1
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T + 1 $ T + 1
$31$	$ T $ T
$37$	$ T + 1 $ T + 1
$41$	$ T - 2 $ T - 2
$43$	$ T $ T
$47$	$ T $ T
$53$	$ T - 2 $ T - 2
$59$	$ T $ T
$61$	$ T + 1 $ T + 1
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T + 1 $ T + 1
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T + 1 $ T + 1
$97$	$ T - 2 $ T - 2
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Elliptic curves over $\Q$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	324.d (of order \(2\), degree \(1\), minimal)

\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(1\)

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