LMFDB - Newform orbit 8400.2.a.t

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8400,2,Mod(1,8400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8400.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:	$67.0743376979$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1050)
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} + q^{9}+O(q^{10}) $

$ q - q^{3} + q^{7} + q^{9} - 2 q^{11} + q^{13} - q^{17} - 4 q^{19} - q^{21} - 7 q^{23} - q^{27} + q^{29} - 3 q^{31} + 2 q^{33} - 6 q^{37} - q^{39} - 3 q^{41} + q^{43} + 12 q^{47} + q^{49} + q^{51} + 11 q^{53} + 4 q^{57} + 3 q^{59} + 5 q^{61} + q^{63} + 12 q^{67} + 7 q^{69} - 4 q^{71} - 14 q^{73} - 2 q^{77} + 2 q^{79} + q^{81} + 3 q^{83} - q^{87} + 10 q^{89} + q^{91} + 3 q^{93} - 10 q^{97} - 2 q^{99}+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

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $
$5$	$ -1 $
$7$	$ -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	8400.2.a.t		1
4.b	odd	2	1		1050.2.a.o	yes	1
5.b	even	2	1		8400.2.a.bt		1
12.b	even	2	1		3150.2.a.c		1
20.d	odd	2	1		1050.2.a.e	✓	1
20.e	even	4	2		1050.2.g.j		2
28.d	even	2	1		7350.2.a.ca		1
60.h	even	2	1		3150.2.a.bl		1
60.l	odd	4	2		3150.2.g.g		2
140.c	even	2	1		7350.2.a.bj		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1050.2.a.e	✓	1	20.d	odd	2	1
1050.2.a.o	yes	1	4.b	odd	2	1
1050.2.g.j		2	20.e	even	4	2
3150.2.a.c		1	12.b	even	2	1
3150.2.a.bl		1	60.h	even	2	1
3150.2.g.g		2	60.l	odd	4	2
7350.2.a.bj		1	140.c	even	2	1
7350.2.a.ca		1	28.d	even	2	1
8400.2.a.t		1	1.a	even	1	1	trivial
8400.2.a.bt		1	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8400))$:

$ T_{11} + 2 $

$ T_{13} - 1 $

$ T_{17} + 1 $

$ T_{19} + 4 $

$ T_{23} + 7 $

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 + 2 $ T + 2
$13$	$ T - 1 $ T - 1
$17$	$ T + 1 $ T + 1
$19$	$ T + 4 $ T + 4
$23$	$ T + 7 $ T + 7
$29$	$ T - 1 $ T - 1
$31$	$ T + 3 $ T + 3
$37$	$ T + 6 $ T + 6
$41$	$ T + 3 $ T + 3
$43$	$ T - 1 $ T - 1
$47$	$ T - 12 $ T - 12
$53$	$ T - 11 $ T - 11
$59$	$ T - 3 $ T - 3
$61$	$ T - 5 $ T - 5
$67$	$ T - 12 $ T - 12
$71$	$ T + 4 $ T + 4
$73$	$ T + 14 $ T + 14
$79$	$ T - 2 $ T - 2
$83$	$ T - 3 $ T - 3
$89$	$ T - 10 $ T - 10
$97$	$ T + 10 $ T + 10
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

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