LMFDB - Newform orbit 900.4.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,4,Mod(1,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, 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("900.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	900.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:	$53.1017190052$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 180)
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 - 2 q^{7}+O(q^{10}) $

$ q - 2 q^{7} - 30 q^{11} + 4 q^{13} + 90 q^{17} - 28 q^{19} + 120 q^{23} - 210 q^{29} - 4 q^{31} - 200 q^{37} - 240 q^{41} + 136 q^{43} - 120 q^{47} - 339 q^{49} - 30 q^{53} + 450 q^{59} - 166 q^{61} - 908 q^{67} + 1020 q^{71} + 250 q^{73} + 60 q^{77} - 916 q^{79} - 1140 q^{83} + 420 q^{89} - 8 q^{91} - 1538 q^{97}+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

−2.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$5$	$1$

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	900.4.a.i		1
3.b	odd	2	1		900.4.a.j		1
5.b	even	2	1		180.4.a.b	✓	1
5.c	odd	4	2		900.4.d.d		2
15.d	odd	2	1		180.4.a.e	yes	1
15.e	even	4	2		900.4.d.i		2
20.d	odd	2	1		720.4.a.h		1
45.h	odd	6	2		1620.4.i.c		2
45.j	even	6	2		1620.4.i.i		2
60.h	even	2	1		720.4.a.w		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
180.4.a.b	✓	1	5.b	even	2	1
180.4.a.e	yes	1	15.d	odd	2	1
720.4.a.h		1	20.d	odd	2	1
720.4.a.w		1	60.h	even	2	1
900.4.a.i		1	1.a	even	1	1	trivial
900.4.a.j		1	3.b	odd	2	1
900.4.d.d		2	5.c	odd	4	2
900.4.d.i		2	15.e	even	4	2
1620.4.i.c		2	45.h	odd	6	2
1620.4.i.i		2	45.j	even	6	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(900))$:

$ T_{7} + 2 $

$ T_{11} + 30 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T + 2 $ T + 2
$11$	$ T + 30 $ T + 30
$13$	$ T - 4 $ T - 4
$17$	$ T - 90 $ T - 90
$19$	$ T + 28 $ T + 28
$23$	$ T - 120 $ T - 120
$29$	$ T + 210 $ T + 210
$31$	$ T + 4 $ T + 4
$37$	$ T + 200 $ T + 200
$41$	$ T + 240 $ T + 240
$43$	$ T - 136 $ T - 136
$47$	$ T + 120 $ T + 120
$53$	$ T + 30 $ T + 30
$59$	$ T - 450 $ T - 450
$61$	$ T + 166 $ T + 166
$67$	$ T + 908 $ T + 908
$71$	$ T - 1020 $ T - 1020
$73$	$ T - 250 $ T - 250
$79$	$ T + 916 $ T + 916
$83$	$ T + 1140 $ T + 1140
$89$	$ T - 420 $ T - 420
$97$	$ T + 1538 $ T + 1538
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	900.a (trivial)

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