LMFDB - Newform orbit 3475.1.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3475,1,Mod(2501,3475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3475.2501");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3475 = 5^{2} \cdot 139 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3475.c (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:	$1.73425091890$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 139)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.139.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.2415125.2

$q$-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^{4} + q^{7} + q^{9}+O(q^{10}) $

$ q + q^{4} + q^{7} + q^{9} - q^{11} + q^{13} + q^{16} + q^{28} - q^{29} - q^{31} + q^{36} - 2 q^{37} + 2 q^{41} - q^{44} - 2 q^{47} + q^{52} + q^{63} + q^{64} + q^{67} - q^{71} - q^{77} - q^{79} + q^{81} + q^{83} - q^{89} + q^{91} - q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3475\mathbb{Z}\right)^\times$.

$n$	$1252$	$2226$
$\chi(n)$	$1$	$-1$

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} $

2501.1

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
139.b	odd	2	1	CM by $\Q(\sqrt{-139}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3475.1.c.b		1
5.b	even	2	1		139.1.b.a	✓	1
5.c	odd	4	2		3475.1.d.a		2
15.d	odd	2	1		1251.1.c.b		1
20.d	odd	2	1		2224.1.e.a		1
139.b	odd	2	1	CM	3475.1.c.b		1
695.d	odd	2	1		139.1.b.a	✓	1
695.g	even	4	2		3475.1.d.a		2
2085.b	even	2	1		1251.1.c.b		1
2780.d	even	2	1		2224.1.e.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
139.1.b.a	✓	1	5.b	even	2	1
139.1.b.a	✓	1	695.d	odd	2	1
1251.1.c.b		1	15.d	odd	2	1
1251.1.c.b		1	2085.b	even	2	1
2224.1.e.a		1	20.d	odd	2	1
2224.1.e.a		1	2780.d	even	2	1
3475.1.c.b		1	1.a	even	1	1	trivial
3475.1.c.b		1	139.b	odd	2	1	CM
3475.1.d.a		2	5.c	odd	4	2
3475.1.d.a		2	695.g	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3475, [\chi])$:

$ T_{2} $

$ T_{7} - 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 1 $ T - 1
$11$	$ T + 1 $ T + 1
$13$	$ T - 1 $ T - 1
$17$	$ T $ T
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T + 1 $ T + 1
$31$	$ T + 1 $ T + 1
$37$	$ T + 2 $ T + 2
$41$	$ T - 2 $ T - 2
$43$	$ T $ T
$47$	$ T + 2 $ T + 2
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T - 1 $ T - 1
$71$	$ T + 1 $ T + 1
$73$	$ T $ T
$79$	$ T + 1 $ T + 1
$83$	$ T - 1 $ T - 1
$89$	$ T + 1 $ T + 1
$97$	$ T $ T
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 3475 = 5^{2} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3475.c (of order \(2\), degree \(1\), minimal)

\(n\)	\(1252\)	\(2226\)
\(\chi(n)\)	\(1\)	\(-1\)

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