LMFDB - Newform orbit 6300.2.a.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6300,2,Mod(1,6300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6300.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.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:	$50.3057532734$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 420)
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^{7}+O(q^{10}) $

$ q - q^{7} + 2 q^{11} - 4 q^{13} + 2 q^{17} + 2 q^{19} + 4 q^{23} - 6 q^{29} - 2 q^{31} - 10 q^{37} + 10 q^{41} - 12 q^{43} - 8 q^{47} + q^{49} + 8 q^{59} - 2 q^{61} + 12 q^{67} + 10 q^{71} - 4 q^{73} - 2 q^{77} - 12 q^{83} - 2 q^{89} + 4 q^{91} + 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

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	6300.2.a.l		1
3.b	odd	2	1		2100.2.a.k		1
5.b	even	2	1		1260.2.a.e		1
5.c	odd	4	2		6300.2.k.m		2
12.b	even	2	1		8400.2.a.bc		1
15.d	odd	2	1		420.2.a.b	✓	1
15.e	even	4	2		2100.2.k.c		2
20.d	odd	2	1		5040.2.a.c		1
35.c	odd	2	1		8820.2.a.x		1
60.h	even	2	1		1680.2.a.r		1
105.g	even	2	1		2940.2.a.g		1
105.o	odd	6	2		2940.2.q.k		2
105.p	even	6	2		2940.2.q.g		2
120.i	odd	2	1		6720.2.a.bt		1
120.m	even	2	1		6720.2.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
420.2.a.b	✓	1	15.d	odd	2	1
1260.2.a.e		1	5.b	even	2	1
1680.2.a.r		1	60.h	even	2	1
2100.2.a.k		1	3.b	odd	2	1
2100.2.k.c		2	15.e	even	4	2
2940.2.a.g		1	105.g	even	2	1
2940.2.q.g		2	105.p	even	6	2
2940.2.q.k		2	105.o	odd	6	2
5040.2.a.c		1	20.d	odd	2	1
6300.2.a.l		1	1.a	even	1	1	trivial
6300.2.k.m		2	5.c	odd	4	2
6720.2.a.d		1	120.m	even	2	1
6720.2.a.bt		1	120.i	odd	2	1
8400.2.a.bc		1	12.b	even	2	1
8820.2.a.x		1	35.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 $S_{2}^{\mathrm{new}}(\Gamma_0(6300))$:

$ T_{11} - 2 $

$ T_{13} + 4 $

$ T_{17} - 2 $

$ T_{37} + 10 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 1 $ T + 1
$11$	$ T - 2 $ T - 2
$13$	$ T + 4 $ T + 4
$17$	$ T - 2 $ T - 2
$19$	$ T - 2 $ T - 2
$23$	$ T - 4 $ T - 4
$29$	$ T + 6 $ T + 6
$31$	$ T + 2 $ T + 2
$37$	$ T + 10 $ T + 10
$41$	$ T - 10 $ T - 10
$43$	$ T + 12 $ T + 12
$47$	$ T + 8 $ T + 8
$53$	$ T $ T
$59$	$ T - 8 $ T - 8
$61$	$ T + 2 $ T + 2
$67$	$ T - 12 $ T - 12
$71$	$ T - 10 $ T - 10
$73$	$ T + 4 $ T + 4
$79$	$ T $ T
$83$	$ T + 12 $ T + 12
$89$	$ T + 2 $ T + 2
$97$	$ T - 8 $ T - 8
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.a (trivial)

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