LMFDB - Newform orbit 336.6.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
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 - 9 q^{3} - 72 q^{5} - 49 q^{7} + 81 q^{9}+O(q^{10}) $

$ q - 9 q^{3} - 72 q^{5} - 49 q^{7} + 81 q^{9} + 414 q^{11} - 1054 q^{13} + 648 q^{15} - 1848 q^{17} - 236 q^{19} + 441 q^{21} - 2898 q^{23} + 2059 q^{25} - 729 q^{27} - 6522 q^{29} - 6200 q^{31} - 3726 q^{33} + 3528 q^{35} + 9650 q^{37} + 9486 q^{39} + 8484 q^{41} + 10804 q^{43} - 5832 q^{45} - 60 q^{47} + 2401 q^{49} + 16632 q^{51} + 22506 q^{53} - 29808 q^{55} + 2124 q^{57} + 28176 q^{59} - 35194 q^{61} - 3969 q^{63} + 75888 q^{65} + 28216 q^{67} + 26082 q^{69} + 6642 q^{71} - 52090 q^{73} - 18531 q^{75} - 20286 q^{77} - 43340 q^{79} + 6561 q^{81} - 25716 q^{83} + 133056 q^{85} + 58698 q^{87} + 98724 q^{89} + 51646 q^{91} + 55800 q^{93} + 16992 q^{95} - 148954 q^{97} + 33534 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

−9.00000

−72.0000

−49.0000

81.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$7$	$1$

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	336.6.a.b		1
3.b	odd	2	1		1008.6.a.ba		1
4.b	odd	2	1		42.6.a.c	✓	1
12.b	even	2	1		126.6.a.l		1
20.d	odd	2	1		1050.6.a.g		1
20.e	even	4	2		1050.6.g.b		2
28.d	even	2	1		294.6.a.c		1
28.f	even	6	2		294.6.e.n		2
28.g	odd	6	2		294.6.e.l		2
84.h	odd	2	1		882.6.a.n		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.c	✓	1	4.b	odd	2	1
126.6.a.l		1	12.b	even	2	1
294.6.a.c		1	28.d	even	2	1
294.6.e.l		2	28.g	odd	6	2
294.6.e.n		2	28.f	even	6	2
336.6.a.b		1	1.a	even	1	1	trivial
882.6.a.n		1	84.h	odd	2	1
1008.6.a.ba		1	3.b	odd	2	1
1050.6.a.g		1	20.d	odd	2	1
1050.6.g.b		2	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(336))$:

$ T_{5} + 72 $

$ T_{11} - 414 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 9 $ T + 9
$5$	$ T + 72 $ T + 72
$7$	$ T + 49 $ T + 49
$11$	$ T - 414 $ T - 414
$13$	$ T + 1054 $ T + 1054
$17$	$ T + 1848 $ T + 1848
$19$	$ T + 236 $ T + 236
$23$	$ T + 2898 $ T + 2898
$29$	$ T + 6522 $ T + 6522
$31$	$ T + 6200 $ T + 6200
$37$	$ T - 9650 $ T - 9650
$41$	$ T - 8484 $ T - 8484
$43$	$ T - 10804 $ T - 10804
$47$	$ T + 60 $ T + 60
$53$	$ T - 22506 $ T - 22506
$59$	$ T - 28176 $ T - 28176
$61$	$ T + 35194 $ T + 35194
$67$	$ T - 28216 $ T - 28216
$71$	$ T - 6642 $ T - 6642
$73$	$ T + 52090 $ T + 52090
$79$	$ T + 43340 $ T + 43340
$83$	$ T + 25716 $ T + 25716
$89$	$ T - 98724 $ T - 98724
$97$	$ T + 148954 $ T + 148954
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

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