LMFDB - Newform orbit 1600.4.a.bo

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 200)
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} - 2 q^{7} - 2 q^{9}+O(q^{10}) $

$ q + 5 q^{3} - 2 q^{7} - 2 q^{9} + 39 q^{11} - 84 q^{13} - 61 q^{17} + 151 q^{19} - 10 q^{21} + 58 q^{23} - 145 q^{27} - 192 q^{29} + 18 q^{31} + 195 q^{33} + 138 q^{37} - 420 q^{39} + 229 q^{41} - 164 q^{43} + 212 q^{47} - 339 q^{49} - 305 q^{51} - 578 q^{53} + 755 q^{57} - 336 q^{59} - 858 q^{61} + 4 q^{63} - 209 q^{67} + 290 q^{69} + 780 q^{71} - 403 q^{73} - 78 q^{77} + 230 q^{79} - 671 q^{81} - 1293 q^{83} - 960 q^{87} - 1369 q^{89} + 168 q^{91} + 90 q^{93} + 382 q^{97} - 78 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

−2.00000

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	1600.4.a.bo		1
4.b	odd	2	1		1600.4.a.m		1
5.b	even	2	1		1600.4.a.n		1
8.b	even	2	1		400.4.a.f		1
8.d	odd	2	1		200.4.a.h	yes	1
20.d	odd	2	1		1600.4.a.bn		1
24.f	even	2	1		1800.4.a.t		1
40.e	odd	2	1		200.4.a.c	✓	1
40.f	even	2	1		400.4.a.q		1
40.i	odd	4	2		400.4.c.g		2
40.k	even	4	2		200.4.c.d		2
120.m	even	2	1		1800.4.a.p		1
120.q	odd	4	2		1800.4.f.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.4.a.c	✓	1	40.e	odd	2	1
200.4.a.h	yes	1	8.d	odd	2	1
200.4.c.d		2	40.k	even	4	2
400.4.a.f		1	8.b	even	2	1
400.4.a.q		1	40.f	even	2	1
400.4.c.g		2	40.i	odd	4	2
1600.4.a.m		1	4.b	odd	2	1
1600.4.a.n		1	5.b	even	2	1
1600.4.a.bn		1	20.d	odd	2	1
1600.4.a.bo		1	1.a	even	1	1	trivial
1800.4.a.p		1	120.m	even	2	1
1800.4.a.t		1	24.f	even	2	1
1800.4.f.c		2	120.q	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1600))$:

$ T_{3} - 5 $

$ T_{7} + 2 $

$ T_{11} - 39 $

$ T_{13} + 84 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 5 $ T - 5
$5$	$ T $ T
$7$	$ T + 2 $ T + 2
$11$	$ T - 39 $ T - 39
$13$	$ T + 84 $ T + 84
$17$	$ T + 61 $ T + 61
$19$	$ T - 151 $ T - 151
$23$	$ T - 58 $ T - 58
$29$	$ T + 192 $ T + 192
$31$	$ T - 18 $ T - 18
$37$	$ T - 138 $ T - 138
$41$	$ T - 229 $ T - 229
$43$	$ T + 164 $ T + 164
$47$	$ T - 212 $ T - 212
$53$	$ T + 578 $ T + 578
$59$	$ T + 336 $ T + 336
$61$	$ T + 858 $ T + 858
$67$	$ T + 209 $ T + 209
$71$	$ T - 780 $ T - 780
$73$	$ T + 403 $ T + 403
$79$	$ T - 230 $ T - 230
$83$	$ T + 1293 $ T + 1293
$89$	$ T + 1369 $ T + 1369
$97$	$ T - 382 $ T - 382
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Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

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