LMFDB - Newform orbit 144.16.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,16,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, 16, names="a")

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

chi := DirichletCharacter("144.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 144 = 2^{4} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 16 $
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:	$205.478647344$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1)
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 - 52110 q^{5} - 2822456 q^{7}+O(q^{10}) $

$ q - 52110 q^{5} - 2822456 q^{7} + 20586852 q^{11} - 190073338 q^{13} - 1646527986 q^{17} - 1563257180 q^{19} + 9451116072 q^{23} - 27802126025 q^{25} + 36902568330 q^{29} - 71588483552 q^{31} + 147078182160 q^{35} - 1033652081554 q^{37} - 1641974018202 q^{41} + 492403109308 q^{43} - 3410684952624 q^{47} + 3218696361993 q^{49} - 6797151655902 q^{53} - 1072780857720 q^{55} + 9858856815540 q^{59} + 4931842626902 q^{61} + 9904721643180 q^{65} + 28837826625364 q^{67} + 125050114914552 q^{71} - 82171455513478 q^{73} - 58105483948512 q^{77} + 25413078694480 q^{79} - 281736730890468 q^{83} + 85800573350460 q^{85} - 715618564776810 q^{89} + 536473633278128 q^{91} + 81461331649800 q^{95} + 612786136081826 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

−52110.0

−2.82246e6

Atkin-Lehner signs

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

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	144.16.a.f		1
3.b	odd	2	1		16.16.a.d		1
4.b	odd	2	1		9.16.a.a		1
12.b	even	2	1		1.16.a.a	✓	1
24.f	even	2	1		64.16.a.i		1
24.h	odd	2	1		64.16.a.c		1
60.h	even	2	1		25.16.a.a		1
60.l	odd	4	2		25.16.b.a		2
84.h	odd	2	1		49.16.a.a		1
84.j	odd	6	2		49.16.c.b		2
84.n	even	6	2		49.16.c.c		2
132.d	odd	2	1		121.16.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1.16.a.a	✓	1	12.b	even	2	1
9.16.a.a		1	4.b	odd	2	1
16.16.a.d		1	3.b	odd	2	1
25.16.a.a		1	60.h	even	2	1
25.16.b.a		2	60.l	odd	4	2
49.16.a.a		1	84.h	odd	2	1
49.16.c.b		2	84.j	odd	6	2
49.16.c.c		2	84.n	even	6	2
64.16.a.c		1	24.h	odd	2	1
64.16.a.i		1	24.f	even	2	1
121.16.a.a		1	132.d	odd	2	1
144.16.a.f		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} + 52110 $ T5 + 52110 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(144))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T $ T
$5$	$ T + 52110 $ T + 52110
$7$	$ T + 2822456 $ T + 2822456
$11$	$ T - 20586852 $ T - 20586852
$13$	$ T + 190073338 $ T + 190073338
$17$	$ T + 1646527986 $ T + 1646527986
$19$	$ T + 1563257180 $ T + 1563257180
$23$	$ T - 9451116072 $ T - 9451116072
$29$	$ T - 36902568330 $ T - 36902568330
$31$	$ T + 71588483552 $ T + 71588483552
$37$	$ T + 1033652081554 $ T + 1033652081554
$41$	$ T + 1641974018202 $ T + 1641974018202
$43$	$ T - 492403109308 $ T - 492403109308
$47$	$ T + 3410684952624 $ T + 3410684952624
$53$	$ T + 6797151655902 $ T + 6797151655902
$59$	$ T - 9858856815540 $ T - 9858856815540
$61$	$ T - 4931842626902 $ T - 4931842626902
$67$	$ T - 28837826625364 $ T - 28837826625364
$71$	$ T - 125050114914552 $ T - 125050114914552
$73$	$ T + 82171455513478 $ T + 82171455513478
$79$	$ T - 25413078694480 $ T - 25413078694480
$83$	$ T + 281736730890468 $ T + 281736730890468
$89$	$ T + 715618564776810 $ T + 715618564776810
$97$	$ T - 612786136081826 $ T - 612786136081826
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Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	144.a (trivial)

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