LMFDB - Newform orbit 1700.2.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1700,2,Mod(1,1700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1700.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1700 = 2^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1700.a (trivial)

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:	$13.5745683436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

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

$ q - q^{3} + q^{7} - 2 q^{9} + q^{13} + q^{17} - 4 q^{19} - q^{21} + 5 q^{27} - 6 q^{29} - q^{31} - 2 q^{37} - q^{39} - 2 q^{43} + 6 q^{47} - 6 q^{49} - q^{51} - 3 q^{53} + 4 q^{57} - 6 q^{59} - 10 q^{61} - 2 q^{63} + 4 q^{67} - 3 q^{71} - 2 q^{73} - q^{79} + q^{81} - 12 q^{83} + 6 q^{87} - 6 q^{89} + q^{91} + q^{93} - 8 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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} $

1.1

−1.00000

1.00000

−2.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ +1 $
$17$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1700.2.a.a	✓	1
4.b	odd	2	1		6800.2.a.p		1
5.b	even	2	1		1700.2.a.c	yes	1
5.c	odd	4	2		1700.2.e.a		2
20.d	odd	2	1		6800.2.a.h		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1700.2.a.a	✓	1	1.a	even	1	1	trivial
1700.2.a.c	yes	1	5.b	even	2	1
1700.2.e.a		2	5.c	odd	4	2
6800.2.a.h		1	20.d	odd	2	1
6800.2.a.p		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 1 $ T3 + 1 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1700))$.

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 - 1 $ T - 1
$17$	$ T - 1 $ T - 1
$19$	$ T + 4 $ T + 4
$23$	$ T $ T
$29$	$ T + 6 $ T + 6
$31$	$ T + 1 $ T + 1
$37$	$ T + 2 $ T + 2
$41$	$ T $ T
$43$	$ T + 2 $ T + 2
$47$	$ T - 6 $ T - 6
$53$	$ T + 3 $ T + 3
$59$	$ T + 6 $ T + 6
$61$	$ T + 10 $ T + 10
$67$	$ T - 4 $ T - 4
$71$	$ T + 3 $ T + 3
$73$	$ T + 2 $ T + 2
$79$	$ T + 1 $ T + 1
$83$	$ T + 12 $ T + 12
$89$	$ T + 6 $ T + 6
$97$	$ T + 8 $ T + 8
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Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1700.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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