LMFDB - Newform orbit 144.10.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,10,Mod(1,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 10, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	144.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:	$74.1651604076$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 9)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$ q + 12580 q^{7}+O(q^{10}) $

$ q + 12580 q^{7} + 118370 q^{13} + 976696 q^{19} - 1953125 q^{25} - 1691228 q^{31} - 15384490 q^{37} + 16577080 q^{43} + 117902793 q^{49} - 117903058 q^{61} - 112542320 q^{67} + 296368310 q^{73} + 616732324 q^{79} + 1489094600 q^{91} + 1288928270 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

12580.0

Atkin-Lehner signs

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

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	144.10.a.h		1
3.b	odd	2	1	CM	144.10.a.h		1
4.b	odd	2	1		9.10.a.b	✓	1
12.b	even	2	1		9.10.a.b	✓	1
20.d	odd	2	1		225.10.a.d		1
20.e	even	4	2		225.10.b.f		2
36.f	odd	6	2		81.10.c.c		2
36.h	even	6	2		81.10.c.c		2
60.h	even	2	1		225.10.a.d		1
60.l	odd	4	2		225.10.b.f		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.10.a.b	✓	1	4.b	odd	2	1
9.10.a.b	✓	1	12.b	even	2	1
81.10.c.c		2	36.f	odd	6	2
81.10.c.c		2	36.h	even	6	2
144.10.a.h		1	1.a	even	1	1	trivial
144.10.a.h		1	3.b	odd	2	1	CM
225.10.a.d		1	20.d	odd	2	1
225.10.a.d		1	60.h	even	2	1
225.10.b.f		2	20.e	even	4	2
225.10.b.f		2	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(144))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T $ T
$7$	$ T - 12580 $ T - 12580
$11$	$ T $ T
$13$	$ T - 118370 $ T - 118370
$17$	$ T $ T
$19$	$ T - 976696 $ T - 976696
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T + 1691228 $ T + 1691228
$37$	$ T + 15384490 $ T + 15384490
$41$	$ T $ T
$43$	$ T - 16577080 $ T - 16577080
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T + 117903058 $ T + 117903058
$67$	$ T + 112542320 $ T + 112542320
$71$	$ T $ T
$73$	$ T - 296368310 $ T - 296368310
$79$	$ T - 616732324 $ T - 616732324
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T - 1288928270 $ T - 1288928270
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

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