LMFDB - Newform orbit 400.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 100)
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 - q^{3} - 26 q^{7} - 26 q^{9}+O(q^{10}) $

$ q - q^{3} - 26 q^{7} - 26 q^{9} - 45 q^{11} + 44 q^{13} + 117 q^{17} + 91 q^{19} + 26 q^{21} + 18 q^{23} + 53 q^{27} + 144 q^{29} - 26 q^{31} + 45 q^{33} - 214 q^{37} - 44 q^{39} - 459 q^{41} + 460 q^{43} + 468 q^{47} + 333 q^{49} - 117 q^{51} + 558 q^{53} - 91 q^{57} + 72 q^{59} - 118 q^{61} + 676 q^{63} - 251 q^{67} - 18 q^{69} - 108 q^{71} + 299 q^{73} + 1170 q^{77} + 898 q^{79} + 649 q^{81} - 927 q^{83} - 144 q^{87} + 351 q^{89} - 1144 q^{91} + 26 q^{93} + 386 q^{97} + 1170 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

−1.00000

−26.0000

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	400.4.a.i		1
4.b	odd	2	1		100.4.a.c	yes	1
5.b	even	2	1		400.4.a.l		1
5.c	odd	4	2		400.4.c.l		2
8.b	even	2	1		1600.4.a.bd		1
8.d	odd	2	1		1600.4.a.x		1
12.b	even	2	1		900.4.a.p		1
20.d	odd	2	1		100.4.a.b	✓	1
20.e	even	4	2		100.4.c.b		2
40.e	odd	2	1		1600.4.a.bc		1
40.f	even	2	1		1600.4.a.y		1
60.h	even	2	1		900.4.a.c		1
60.l	odd	4	2		900.4.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
100.4.a.b	✓	1	20.d	odd	2	1
100.4.a.c	yes	1	4.b	odd	2	1
100.4.c.b		2	20.e	even	4	2
400.4.a.i		1	1.a	even	1	1	trivial
400.4.a.l		1	5.b	even	2	1
400.4.c.l		2	5.c	odd	4	2
900.4.a.c		1	60.h	even	2	1
900.4.a.p		1	12.b	even	2	1
900.4.d.a		2	60.l	odd	4	2
1600.4.a.x		1	8.d	odd	2	1
1600.4.a.y		1	40.f	even	2	1
1600.4.a.bc		1	40.e	odd	2	1
1600.4.a.bd		1	8.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(400))$:

$ T_{3} + 1 $

$ T_{7} + 26 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 1 $ T + 1
$5$	$ T $ T
$7$	$ T + 26 $ T + 26
$11$	$ T + 45 $ T + 45
$13$	$ T - 44 $ T - 44
$17$	$ T - 117 $ T - 117
$19$	$ T - 91 $ T - 91
$23$	$ T - 18 $ T - 18
$29$	$ T - 144 $ T - 144
$31$	$ T + 26 $ T + 26
$37$	$ T + 214 $ T + 214
$41$	$ T + 459 $ T + 459
$43$	$ T - 460 $ T - 460
$47$	$ T - 468 $ T - 468
$53$	$ T - 558 $ T - 558
$59$	$ T - 72 $ T - 72
$61$	$ T + 118 $ T + 118
$67$	$ T + 251 $ T + 251
$71$	$ T + 108 $ T + 108
$73$	$ T - 299 $ T - 299
$79$	$ T - 898 $ T - 898
$83$	$ T + 927 $ T + 927
$89$	$ T - 351 $ T - 351
$97$	$ T - 386 $ T - 386
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Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

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