LMFDB - Newform orbit 3248.1.k.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3248,1,Mod(1217,3248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3248, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3248.1217");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3248 = 2^{4} \cdot 7 \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3248.k (of order $2$, degree $1$, not 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:	$1.62096316103$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 812)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.812.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.73846528.1

$q$-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^{3} - q^{7}+O(q^{10}) $

$ q - q^{3} - q^{7} - 2 q^{17} - q^{19} + q^{21} + q^{23} + q^{25} + q^{27} + q^{29} + 2 q^{31} + q^{41} - q^{47} + q^{49} + 2 q^{51} - q^{53} + q^{57} - 2 q^{61} + q^{67} - q^{69} + q^{71} + q^{73} - q^{75} - q^{81} - q^{87} + q^{89} - 2 q^{93} + q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3248\mathbb{Z}\right)^\times$.

$n$	$465$	$785$	$2031$	$2437$
$\chi(n)$	$1$	$1$	$0$	$0$

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} $

1217.1

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
203.c	odd	2	1	CM by $\Q(\sqrt{-203}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3248.1.k.a		1
4.b	odd	2	1		812.1.g.b	yes	1
7.b	odd	2	1		3248.1.k.c		1
28.d	even	2	1		812.1.g.a	✓	1
29.b	even	2	1		3248.1.k.c		1
116.d	odd	2	1		812.1.g.a	✓	1
203.c	odd	2	1	CM	3248.1.k.a		1
812.c	even	2	1		812.1.g.b	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
812.1.g.a	✓	1	28.d	even	2	1
812.1.g.a	✓	1	116.d	odd	2	1
812.1.g.b	yes	1	4.b	odd	2	1
812.1.g.b	yes	1	812.c	even	2	1
3248.1.k.a		1	1.a	even	1	1	trivial
3248.1.k.a		1	203.c	odd	2	1	CM
3248.1.k.c		1	7.b	odd	2	1
3248.1.k.c		1	29.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 1 $ T3 + 1 acting on $S_{1}^{\mathrm{new}}(3248, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T + 1 $ T + 1
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T + 2 $ T + 2
$19$	$ T + 1 $ T + 1
$23$	$ T - 1 $ T - 1
$29$	$ T - 1 $ T - 1
$31$	$ T - 2 $ T - 2
$37$	$ T $ T
$41$	$ T - 1 $ T - 1
$43$	$ T $ T
$47$	$ T + 1 $ T + 1
$53$	$ T + 1 $ T + 1
$59$	$ T $ T
$61$	$ T + 2 $ T + 2
$67$	$ T - 1 $ T - 1
$71$	$ T - 1 $ T - 1
$73$	$ T - 1 $ T - 1
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T - 1 $ T - 1
$97$	$ T - 1 $ T - 1
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Level:	\( N \)	\(=\)	\( 3248 = 2^{4} \cdot 7 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3248.k (of order \(2\), degree \(1\), not minimal)

\(n\)	\(465\)	\(785\)	\(2031\)	\(2437\)
\(\chi(n)\)	\(1\)	\(1\)	\(0\)	\(0\)

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