LMFDB - Newform orbit 2450.4.a.bf

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,4,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2450.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:	$144.554679514$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 14)
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 + 2 q^{2} - q^{3} + 4 q^{4} - 2 q^{6} + 8 q^{8} - 26 q^{9} + 35 q^{11} - 4 q^{12} + 66 q^{13} + 16 q^{16} + 59 q^{17} - 52 q^{18} - 137 q^{19} + 70 q^{22} + 7 q^{23} - 8 q^{24} + 132 q^{26} + 53 q^{27}+ \cdots - 910 q^{99}+O(q^{100}) $

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

2.00000

−1.00000

4.00000

−2.00000

8.00000

−26.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -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	2450.4.a.bf		1
5.b	even	2	1		98.4.a.c		1
7.b	odd	2	1		2450.4.a.bh		1
7.d	odd	6	2		350.4.e.b		2
15.d	odd	2	1		882.4.a.p		1
20.d	odd	2	1		784.4.a.j		1
35.c	odd	2	1		98.4.a.b		1
35.i	odd	6	2		14.4.c.b	✓	2
35.j	even	6	2		98.4.c.e		2
35.k	even	12	4		350.4.j.d		4
105.g	even	2	1		882.4.a.k		1
105.o	odd	6	2		882.4.g.d		2
105.p	even	6	2		126.4.g.c		2
140.c	even	2	1		784.4.a.l		1
140.s	even	6	2		112.4.i.b		2
280.ba	even	6	2		448.4.i.d		2
280.bk	odd	6	2		448.4.i.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
14.4.c.b	✓	2	35.i	odd	6	2
98.4.a.b		1	35.c	odd	2	1
98.4.a.c		1	5.b	even	2	1
98.4.c.e		2	35.j	even	6	2
112.4.i.b		2	140.s	even	6	2
126.4.g.c		2	105.p	even	6	2
350.4.e.b		2	7.d	odd	6	2
350.4.j.d		4	35.k	even	12	4
448.4.i.c		2	280.bk	odd	6	2
448.4.i.d		2	280.ba	even	6	2
784.4.a.j		1	20.d	odd	2	1
784.4.a.l		1	140.c	even	2	1
882.4.a.k		1	105.g	even	2	1
882.4.a.p		1	15.d	odd	2	1
882.4.g.d		2	105.o	odd	6	2
2450.4.a.bf		1	1.a	even	1	1	trivial
2450.4.a.bh		1	7.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2450))$:

$ T_{3} + 1 $

$ T_{11} - 35 $

$ T_{19} + 137 $

$ T_{23} - 7 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 2 $ T - 2
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T $ T
$11$	$ T - 35 $ T - 35
$13$	$ T - 66 $ T - 66
$17$	$ T - 59 $ T - 59
$19$	$ T + 137 $ T + 137
$23$	$ T - 7 $ T - 7
$29$	$ T - 106 $ T - 106
$31$	$ T + 75 $ T + 75
$37$	$ T + 11 $ T + 11
$41$	$ T - 498 $ T - 498
$43$	$ T + 260 $ T + 260
$47$	$ T + 171 $ T + 171
$53$	$ T - 417 $ T - 417
$59$	$ T - 17 $ T - 17
$61$	$ T + 51 $ T + 51
$67$	$ T + 439 $ T + 439
$71$	$ T + 784 $ T + 784
$73$	$ T - 295 $ T - 295
$79$	$ T + 495 $ T + 495
$83$	$ T - 932 $ T - 932
$89$	$ T - 873 $ T - 873
$97$	$ T + 290 $ T + 290
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Level:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

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