LMFDB - Newform orbit 3575.1.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3575,1,Mod(2001,3575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3575.2001");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3575 = 5^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3575.h (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:	$1.78415742016$
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 3.1.3575.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.0.702934375.1

$q$-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} - 2 q^{3} + 2 q^{6} - q^{7} + q^{8} + 3 q^{9}+O(q^{10}) $

$ q - q^{2} - 2 q^{3} + 2 q^{6} - q^{7} + q^{8} + 3 q^{9} - q^{11} + q^{13} + q^{14} - q^{16} - 3 q^{18} + q^{19} + 2 q^{21} + q^{22} + q^{23} - 2 q^{24} - q^{26} - 4 q^{27} + 2 q^{33} - q^{38} - 2 q^{39} - 2 q^{41} - 2 q^{42} - q^{46} + 2 q^{48} + q^{53} + 4 q^{54} - q^{56} - 2 q^{57} - 3 q^{63} + q^{64} - 2 q^{66} - 2 q^{69} + 3 q^{72} - q^{73} + q^{77} + 2 q^{78} + 5 q^{81} + 2 q^{82} - q^{83} - q^{88} - q^{91} - 3 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3575\mathbb{Z}\right)^\times$.

$n$	$651$	$1002$	$1926$
$\chi(n)$	$-1$	$1$	$-1$

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} $

2001.1

−1.00000

−2.00000

2.00000

−1.00000

1.00000

3.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
143.d	odd	2	1	CM by $\Q(\sqrt{-143}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3575.1.h.a	✓	1
5.b	even	2	1		3575.1.h.d	yes	1
5.c	odd	4	2		3575.1.c.b		2
11.b	odd	2	1		3575.1.h.c	yes	1
13.b	even	2	1		3575.1.h.c	yes	1
55.d	odd	2	1		3575.1.h.b	yes	1
55.e	even	4	2		3575.1.c.a		2
65.d	even	2	1		3575.1.h.b	yes	1
65.h	odd	4	2		3575.1.c.a		2
143.d	odd	2	1	CM	3575.1.h.a	✓	1
715.c	odd	2	1		3575.1.h.d	yes	1
715.q	even	4	2		3575.1.c.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3575.1.c.a		2	55.e	even	4	2
3575.1.c.a		2	65.h	odd	4	2
3575.1.c.b		2	5.c	odd	4	2
3575.1.c.b		2	715.q	even	4	2
3575.1.h.a	✓	1	1.a	even	1	1	trivial
3575.1.h.a	✓	1	143.d	odd	2	1	CM
3575.1.h.b	yes	1	55.d	odd	2	1
3575.1.h.b	yes	1	65.d	even	2	1
3575.1.h.c	yes	1	11.b	odd	2	1
3575.1.h.c	yes	1	13.b	even	2	1
3575.1.h.d	yes	1	5.b	even	2	1
3575.1.h.d	yes	1	715.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3575, [\chi])$:

$ T_{2} + 1 $

$ T_{3} + 2 $

Hecke characteristic polynomials

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

Level:	\( N \)	\(=\)	\( 3575 = 5^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3575.h (of order \(2\), degree \(1\), minimal)

\(n\)	\(651\)	\(1002\)	\(1926\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)

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