LMFDB - Newform orbit 2240.4.a.bc

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
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 + 5 q^{3} - 5 q^{5} + 7 q^{7} - 2 q^{9}+O(q^{10}) $

$ q + 5 q^{3} - 5 q^{5} + 7 q^{7} - 2 q^{9} - q^{11} - 7 q^{13} - 25 q^{15} - 51 q^{17} + 30 q^{19} + 35 q^{21} + 50 q^{23} + 25 q^{25} - 145 q^{27} - 79 q^{29} + 212 q^{31} - 5 q^{33} - 35 q^{35} + 190 q^{37} - 35 q^{39} - 308 q^{41} + 422 q^{43} + 10 q^{45} - 121 q^{47} + 49 q^{49} - 255 q^{51} - 664 q^{53} + 5 q^{55} + 150 q^{57} + 628 q^{59} + 684 q^{61} - 14 q^{63} + 35 q^{65} + 1056 q^{67} + 250 q^{69} - 744 q^{71} + 726 q^{73} + 125 q^{75} - 7 q^{77} + 407 q^{79} - 671 q^{81} + 644 q^{83} + 255 q^{85} - 395 q^{87} - 880 q^{89} - 49 q^{91} + 1060 q^{93} - 150 q^{95} - 1351 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

5.00000

−5.00000

7.00000

−2.00000

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	2240.4.a.bc		1
4.b	odd	2	1		2240.4.a.h		1
8.b	even	2	1		560.4.a.f		1
8.d	odd	2	1		70.4.a.e	✓	1
24.f	even	2	1		630.4.a.b		1
40.e	odd	2	1		350.4.a.c		1
40.k	even	4	2		350.4.c.k		2
56.e	even	2	1		490.4.a.j		1
56.k	odd	6	2		490.4.e.c		2
56.m	even	6	2		490.4.e.g		2
280.n	even	2	1		2450.4.a.r		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.e	✓	1	8.d	odd	2	1
350.4.a.c		1	40.e	odd	2	1
350.4.c.k		2	40.k	even	4	2
490.4.a.j		1	56.e	even	2	1
490.4.e.c		2	56.k	odd	6	2
490.4.e.g		2	56.m	even	6	2
560.4.a.f		1	8.b	even	2	1
630.4.a.b		1	24.f	even	2	1
2240.4.a.h		1	4.b	odd	2	1
2240.4.a.bc		1	1.a	even	1	1	trivial
2450.4.a.r		1	280.n	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2240))$:

$ T_{3} - 5 $

$ T_{11} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 5 $ T - 5
$5$	$ T + 5 $ T + 5
$7$	$ T - 7 $ T - 7
$11$	$ T + 1 $ T + 1
$13$	$ T + 7 $ T + 7
$17$	$ T + 51 $ T + 51
$19$	$ T - 30 $ T - 30
$23$	$ T - 50 $ T - 50
$29$	$ T + 79 $ T + 79
$31$	$ T - 212 $ T - 212
$37$	$ T - 190 $ T - 190
$41$	$ T + 308 $ T + 308
$43$	$ T - 422 $ T - 422
$47$	$ T + 121 $ T + 121
$53$	$ T + 664 $ T + 664
$59$	$ T - 628 $ T - 628
$61$	$ T - 684 $ T - 684
$67$	$ T - 1056 $ T - 1056
$71$	$ T + 744 $ T + 744
$73$	$ T - 726 $ T - 726
$79$	$ T - 407 $ T - 407
$83$	$ T - 644 $ T - 644
$89$	$ T + 880 $ T + 880
$97$	$ T + 1351 $ T + 1351
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Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

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