LMFDB - Newform orbit 2240.4.a.c

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 140)
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 - 8 q^{3} + 5 q^{5} + 7 q^{7} + 37 q^{9}+O(q^{10}) $

$ q - 8 q^{3} + 5 q^{5} + 7 q^{7} + 37 q^{9} - 28 q^{11} - 82 q^{13} - 40 q^{15} - 46 q^{17} - 8 q^{19} - 56 q^{21} - 128 q^{23} + 25 q^{25} - 80 q^{27} - 174 q^{29} - 152 q^{31} + 224 q^{33} + 35 q^{35} + 290 q^{37} + 656 q^{39} + 50 q^{41} - 396 q^{43} + 185 q^{45} - 296 q^{47} + 49 q^{49} + 368 q^{51} + 570 q^{53} - 140 q^{55} + 64 q^{57} + 272 q^{59} + 662 q^{61} + 259 q^{63} - 410 q^{65} - 876 q^{67} + 1024 q^{69} - 880 q^{71} - 638 q^{73} - 200 q^{75} - 196 q^{77} - 600 q^{79} - 359 q^{81} - 624 q^{83} - 230 q^{85} + 1392 q^{87} + 698 q^{89} - 574 q^{91} + 1216 q^{93} - 40 q^{95} + 754 q^{97} - 1036 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

−8.00000

5.00000

7.00000

37.0000

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	2240.4.a.c		1
4.b	odd	2	1		2240.4.a.bj		1
8.b	even	2	1		140.4.a.e	✓	1
8.d	odd	2	1		560.4.a.b		1
24.h	odd	2	1		1260.4.a.j		1
40.f	even	2	1		700.4.a.b		1
40.i	odd	4	2		700.4.e.c		2
56.h	odd	2	1		980.4.a.b		1
56.j	odd	6	2		980.4.i.q		2
56.p	even	6	2		980.4.i.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.4.a.e	✓	1	8.b	even	2	1
560.4.a.b		1	8.d	odd	2	1
700.4.a.b		1	40.f	even	2	1
700.4.e.c		2	40.i	odd	4	2
980.4.a.b		1	56.h	odd	2	1
980.4.i.b		2	56.p	even	6	2
980.4.i.q		2	56.j	odd	6	2
1260.4.a.j		1	24.h	odd	2	1
2240.4.a.c		1	1.a	even	1	1	trivial
2240.4.a.bj		1	4.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 $S_{4}^{\mathrm{new}}(\Gamma_0(2240))$:

$ T_{3} + 8 $

$ T_{11} + 28 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 8 $ T + 8
$5$	$ T - 5 $ T - 5
$7$	$ T - 7 $ T - 7
$11$	$ T + 28 $ T + 28
$13$	$ T + 82 $ T + 82
$17$	$ T + 46 $ T + 46
$19$	$ T + 8 $ T + 8
$23$	$ T + 128 $ T + 128
$29$	$ T + 174 $ T + 174
$31$	$ T + 152 $ T + 152
$37$	$ T - 290 $ T - 290
$41$	$ T - 50 $ T - 50
$43$	$ T + 396 $ T + 396
$47$	$ T + 296 $ T + 296
$53$	$ T - 570 $ T - 570
$59$	$ T - 272 $ T - 272
$61$	$ T - 662 $ T - 662
$67$	$ T + 876 $ T + 876
$71$	$ T + 880 $ T + 880
$73$	$ T + 638 $ T + 638
$79$	$ T + 600 $ T + 600
$83$	$ T + 624 $ T + 624
$89$	$ T - 698 $ T - 698
$97$	$ T - 754 $ T - 754
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Elliptic curves over $\Q$
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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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