LMFDB - Newform orbit 1300.1.e.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1300,1,Mod(51,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, 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("1300.51");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1300.e (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:	$0.648784516423$
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 3.1.1300.1
Artin image:	$S_3$
Artin field:	Galois closure of 3.1.1300.1

$q$-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^{7} + q^{8} + q^{9}+O(q^{10}) $

$ q + q^{2} + q^{4} - q^{7} + q^{8} + q^{9} - q^{11} + q^{13} - q^{14} + q^{16} - q^{17} + q^{18} + 2 q^{19} - q^{22} + q^{26} - q^{28} - q^{29} - q^{31} + q^{32} - q^{34} + q^{36} + 2 q^{38} - q^{44} - q^{47} + q^{52} - q^{53} - q^{56} - q^{58} - q^{59} - q^{61} - q^{62} - q^{63} + q^{64} - q^{67} - q^{68} + 2 q^{71} + q^{72} + 2 q^{76} + q^{77} + q^{81} - q^{83} - q^{88} - q^{91} - q^{94} - q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$-1$	$-1$	$1$

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} $

51.1

1.00000

−1.00000

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
52.b	odd	2	1	CM by $\Q(\sqrt{-13}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1300.1.e.c	yes	1
4.b	odd	2	1		1300.1.e.b	yes	1
5.b	even	2	1		1300.1.e.a	✓	1
5.c	odd	4	2		1300.1.g.a		2
13.b	even	2	1		1300.1.e.b	yes	1
20.d	odd	2	1		1300.1.e.d	yes	1
20.e	even	4	2		1300.1.g.b		2
52.b	odd	2	1	CM	1300.1.e.c	yes	1
65.d	even	2	1		1300.1.e.d	yes	1
65.h	odd	4	2		1300.1.g.b		2
260.g	odd	2	1		1300.1.e.a	✓	1
260.p	even	4	2		1300.1.g.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1300.1.e.a	✓	1	5.b	even	2	1
1300.1.e.a	✓	1	260.g	odd	2	1
1300.1.e.b	yes	1	4.b	odd	2	1
1300.1.e.b	yes	1	13.b	even	2	1
1300.1.e.c	yes	1	1.a	even	1	1	trivial
1300.1.e.c	yes	1	52.b	odd	2	1	CM
1300.1.e.d	yes	1	20.d	odd	2	1
1300.1.e.d	yes	1	65.d	even	2	1
1300.1.g.a		2	5.c	odd	4	2
1300.1.g.a		2	260.p	even	4	2
1300.1.g.b		2	20.e	even	4	2
1300.1.g.b		2	65.h	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1300, [\chi])$:

$ T_{7} + 1 $

$ T_{11} + 1 $

Hecke characteristic polynomials

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

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1300.e (of order \(2\), degree \(1\), minimal)

\(n\)	\(301\)	\(651\)	\(677\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(1\)

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