LMFDB - Newform orbit 1200.4.a.bj

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
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 + 3 q^{3} + 20 q^{7} + 9 q^{9}+O(q^{10}) $

$ q + 3 q^{3} + 20 q^{7} + 9 q^{9} - 16 q^{11} - 58 q^{13} - 38 q^{17} - 4 q^{19} + 60 q^{21} - 80 q^{23} + 27 q^{27} + 82 q^{29} + 8 q^{31} - 48 q^{33} - 426 q^{37} - 174 q^{39} - 246 q^{41} - 524 q^{43} - 464 q^{47} + 57 q^{49} - 114 q^{51} + 702 q^{53} - 12 q^{57} + 592 q^{59} + 574 q^{61} + 180 q^{63} - 172 q^{67} - 240 q^{69} - 768 q^{71} + 558 q^{73} - 320 q^{77} - 408 q^{79} + 81 q^{81} + 164 q^{83} + 246 q^{87} - 510 q^{89} - 1160 q^{91} + 24 q^{93} - 514 q^{97} - 144 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

3.00000

20.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ -1 $
$5$	$ +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	1200.4.a.bj		1
4.b	odd	2	1		600.4.a.a		1
5.b	even	2	1		240.4.a.a		1
5.c	odd	4	2		1200.4.f.h		2
12.b	even	2	1		1800.4.a.e		1
15.d	odd	2	1		720.4.a.s		1
20.d	odd	2	1		120.4.a.e	✓	1
20.e	even	4	2		600.4.f.f		2
40.e	odd	2	1		960.4.a.q		1
40.f	even	2	1		960.4.a.bd		1
60.h	even	2	1		360.4.a.m		1
60.l	odd	4	2		1800.4.f.k		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.a.e	✓	1	20.d	odd	2	1
240.4.a.a		1	5.b	even	2	1
360.4.a.m		1	60.h	even	2	1
600.4.a.a		1	4.b	odd	2	1
600.4.f.f		2	20.e	even	4	2
720.4.a.s		1	15.d	odd	2	1
960.4.a.q		1	40.e	odd	2	1
960.4.a.bd		1	40.f	even	2	1
1200.4.a.bj		1	1.a	even	1	1	trivial
1200.4.f.h		2	5.c	odd	4	2
1800.4.a.e		1	12.b	even	2	1
1800.4.f.k		2	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1200))$:

$ T_{7} - 20 $

$ T_{11} + 16 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 3 $ T - 3
$5$	$ T $ T
$7$	$ T - 20 $ T - 20
$11$	$ T + 16 $ T + 16
$13$	$ T + 58 $ T + 58
$17$	$ T + 38 $ T + 38
$19$	$ T + 4 $ T + 4
$23$	$ T + 80 $ T + 80
$29$	$ T - 82 $ T - 82
$31$	$ T - 8 $ T - 8
$37$	$ T + 426 $ T + 426
$41$	$ T + 246 $ T + 246
$43$	$ T + 524 $ T + 524
$47$	$ T + 464 $ T + 464
$53$	$ T - 702 $ T - 702
$59$	$ T - 592 $ T - 592
$61$	$ T - 574 $ T - 574
$67$	$ T + 172 $ T + 172
$71$	$ T + 768 $ T + 768
$73$	$ T - 558 $ T - 558
$79$	$ T + 408 $ T + 408
$83$	$ T - 164 $ T - 164
$89$	$ T + 510 $ T + 510
$97$	$ T + 514 $ T + 514
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

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