LMFDB - Newform orbit 16.6.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,6,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	16.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:	$2.56614111701$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4)
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 + 12 q^{3} + 54 q^{5} + 88 q^{7} - 99 q^{9}+O(q^{10}) $

$ q + 12 q^{3} + 54 q^{5} + 88 q^{7} - 99 q^{9} - 540 q^{11} - 418 q^{13} + 648 q^{15} + 594 q^{17} - 836 q^{19} + 1056 q^{21} + 4104 q^{23} - 209 q^{25} - 4104 q^{27} - 594 q^{29} - 4256 q^{31} - 6480 q^{33} + 4752 q^{35} - 298 q^{37} - 5016 q^{39} + 17226 q^{41} + 12100 q^{43} - 5346 q^{45} + 1296 q^{47} - 9063 q^{49} + 7128 q^{51} + 19494 q^{53} - 29160 q^{55} - 10032 q^{57} + 7668 q^{59} - 34738 q^{61} - 8712 q^{63} - 22572 q^{65} - 21812 q^{67} + 49248 q^{69} + 46872 q^{71} + 67562 q^{73} - 2508 q^{75} - 47520 q^{77} + 76912 q^{79} - 25191 q^{81} - 67716 q^{83} + 32076 q^{85} - 7128 q^{87} + 29754 q^{89} - 36784 q^{91} - 51072 q^{93} - 45144 q^{95} - 122398 q^{97} + 53460 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

Expression as an eta quotient

$f(z) = \dfrac{\eta(4z)^{36}}{\eta(2z)^{12}\eta(8z)^{12}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-12}(1 - q^{4n})^{36}(1 - q^{8n})^{-12}$

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

12.0000

54.0000

88.0000

−99.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$

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	16.6.a.b		1
3.b	odd	2	1		144.6.a.c		1
4.b	odd	2	1		4.6.a.a	✓	1
5.b	even	2	1		400.6.a.d		1
5.c	odd	4	2		400.6.c.f		2
7.b	odd	2	1		784.6.a.d		1
8.b	even	2	1		64.6.a.b		1
8.d	odd	2	1		64.6.a.f		1
12.b	even	2	1		36.6.a.a		1
16.e	even	4	2		256.6.b.c		2
16.f	odd	4	2		256.6.b.g		2
20.d	odd	2	1		100.6.a.b		1
20.e	even	4	2		100.6.c.b		2
24.f	even	2	1		576.6.a.bc		1
24.h	odd	2	1		576.6.a.bd		1
28.d	even	2	1		196.6.a.e		1
28.f	even	6	2		196.6.e.d		2
28.g	odd	6	2		196.6.e.g		2
36.f	odd	6	2		324.6.e.a		2
36.h	even	6	2		324.6.e.d		2
44.c	even	2	1		484.6.a.a		1
52.b	odd	2	1		676.6.a.a		1
52.f	even	4	2		676.6.d.a		2
60.h	even	2	1		900.6.a.h		1
60.l	odd	4	2		900.6.d.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4.6.a.a	✓	1	4.b	odd	2	1
16.6.a.b		1	1.a	even	1	1	trivial
36.6.a.a		1	12.b	even	2	1
64.6.a.b		1	8.b	even	2	1
64.6.a.f		1	8.d	odd	2	1
100.6.a.b		1	20.d	odd	2	1
100.6.c.b		2	20.e	even	4	2
144.6.a.c		1	3.b	odd	2	1
196.6.a.e		1	28.d	even	2	1
196.6.e.d		2	28.f	even	6	2
196.6.e.g		2	28.g	odd	6	2
256.6.b.c		2	16.e	even	4	2
256.6.b.g		2	16.f	odd	4	2
324.6.e.a		2	36.f	odd	6	2
324.6.e.d		2	36.h	even	6	2
400.6.a.d		1	5.b	even	2	1
400.6.c.f		2	5.c	odd	4	2
484.6.a.a		1	44.c	even	2	1
576.6.a.bc		1	24.f	even	2	1
576.6.a.bd		1	24.h	odd	2	1
676.6.a.a		1	52.b	odd	2	1
676.6.d.a		2	52.f	even	4	2
784.6.a.d		1	7.b	odd	2	1
900.6.a.h		1	60.h	even	2	1
900.6.d.a		2	60.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 12 $ T3 - 12 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(16))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 12 $ T - 12
$5$	$ T - 54 $ T - 54
$7$	$ T - 88 $ T - 88
$11$	$ T + 540 $ T + 540
$13$	$ T + 418 $ T + 418
$17$	$ T - 594 $ T - 594
$19$	$ T + 836 $ T + 836
$23$	$ T - 4104 $ T - 4104
$29$	$ T + 594 $ T + 594
$31$	$ T + 4256 $ T + 4256
$37$	$ T + 298 $ T + 298
$41$	$ T - 17226 $ T - 17226
$43$	$ T - 12100 $ T - 12100
$47$	$ T - 1296 $ T - 1296
$53$	$ T - 19494 $ T - 19494
$59$	$ T - 7668 $ T - 7668
$61$	$ T + 34738 $ T + 34738
$67$	$ T + 21812 $ T + 21812
$71$	$ T - 46872 $ T - 46872
$73$	$ T - 67562 $ T - 67562
$79$	$ T - 76912 $ T - 76912
$83$	$ T + 67716 $ T + 67716
$89$	$ T - 29754 $ T - 29754
$97$	$ T + 122398 $ T + 122398
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

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