LMFDB - Newform orbit 320.4.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,4,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("320.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.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:	$18.8806112018$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 20)
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 + 4 q^{3} - 5 q^{5} + 16 q^{7} - 11 q^{9}+O(q^{10}) $

$ q + 4 q^{3} - 5 q^{5} + 16 q^{7} - 11 q^{9} - 60 q^{11} - 86 q^{13} - 20 q^{15} + 18 q^{17} + 44 q^{19} + 64 q^{21} - 48 q^{23} + 25 q^{25} - 152 q^{27} + 186 q^{29} - 176 q^{31} - 240 q^{33} - 80 q^{35} - 254 q^{37} - 344 q^{39} + 186 q^{41} - 100 q^{43} + 55 q^{45} - 168 q^{47} - 87 q^{49} + 72 q^{51} + 498 q^{53} + 300 q^{55} + 176 q^{57} - 252 q^{59} + 58 q^{61} - 176 q^{63} + 430 q^{65} - 1036 q^{67} - 192 q^{69} - 168 q^{71} + 506 q^{73} + 100 q^{75} - 960 q^{77} - 272 q^{79} - 311 q^{81} + 948 q^{83} - 90 q^{85} + 744 q^{87} - 1014 q^{89} - 1376 q^{91} - 704 q^{93} - 220 q^{95} - 766 q^{97} + 660 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

4.00000

−5.00000

16.0000

−11.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$5$	$1$

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	320.4.a.k		1
4.b	odd	2	1		320.4.a.d		1
5.b	even	2	1		1600.4.a.p		1
8.b	even	2	1		80.4.a.c		1
8.d	odd	2	1		20.4.a.a	✓	1
16.e	even	4	2		1280.4.d.c		2
16.f	odd	4	2		1280.4.d.n		2
20.d	odd	2	1		1600.4.a.bl		1
24.f	even	2	1		180.4.a.a		1
24.h	odd	2	1		720.4.a.k		1
40.e	odd	2	1		100.4.a.a		1
40.f	even	2	1		400.4.a.o		1
40.i	odd	4	2		400.4.c.j		2
40.k	even	4	2		100.4.c.a		2
56.e	even	2	1		980.4.a.c		1
56.k	odd	6	2		980.4.i.e		2
56.m	even	6	2		980.4.i.n		2
72.l	even	6	2		1620.4.i.j		2
72.p	odd	6	2		1620.4.i.d		2
88.g	even	2	1		2420.4.a.d		1
120.m	even	2	1		900.4.a.m		1
120.q	odd	4	2		900.4.d.k		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.4.a.a	✓	1	8.d	odd	2	1
80.4.a.c		1	8.b	even	2	1
100.4.a.a		1	40.e	odd	2	1
100.4.c.a		2	40.k	even	4	2
180.4.a.a		1	24.f	even	2	1
320.4.a.d		1	4.b	odd	2	1
320.4.a.k		1	1.a	even	1	1	trivial
400.4.a.o		1	40.f	even	2	1
400.4.c.j		2	40.i	odd	4	2
720.4.a.k		1	24.h	odd	2	1
900.4.a.m		1	120.m	even	2	1
900.4.d.k		2	120.q	odd	4	2
980.4.a.c		1	56.e	even	2	1
980.4.i.e		2	56.k	odd	6	2
980.4.i.n		2	56.m	even	6	2
1280.4.d.c		2	16.e	even	4	2
1280.4.d.n		2	16.f	odd	4	2
1600.4.a.p		1	5.b	even	2	1
1600.4.a.bl		1	20.d	odd	2	1
1620.4.i.d		2	72.p	odd	6	2
1620.4.i.j		2	72.l	even	6	2
2420.4.a.d		1	88.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(320))$:

$ T_{3} - 4 $

$ T_{7} - 16 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 4 $ T - 4
$5$	$ T + 5 $ T + 5
$7$	$ T - 16 $ T - 16
$11$	$ T + 60 $ T + 60
$13$	$ T + 86 $ T + 86
$17$	$ T - 18 $ T - 18
$19$	$ T - 44 $ T - 44
$23$	$ T + 48 $ T + 48
$29$	$ T - 186 $ T - 186
$31$	$ T + 176 $ T + 176
$37$	$ T + 254 $ T + 254
$41$	$ T - 186 $ T - 186
$43$	$ T + 100 $ T + 100
$47$	$ T + 168 $ T + 168
$53$	$ T - 498 $ T - 498
$59$	$ T + 252 $ T + 252
$61$	$ T - 58 $ T - 58
$67$	$ T + 1036 $ T + 1036
$71$	$ T + 168 $ T + 168
$73$	$ T - 506 $ T - 506
$79$	$ T + 272 $ T + 272
$83$	$ T - 948 $ T - 948
$89$	$ T + 1014 $ T + 1014
$97$	$ T + 766 $ T + 766
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

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