LMFDB - Newform orbit 336.6.a.i

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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
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} + 94 q^{5} + 49 q^{7} + 81 q^{9}+O(q^{10}) $

$ q - 9 q^{3} + 94 q^{5} + 49 q^{7} + 81 q^{9} - 52 q^{11} - 770 q^{13} - 846 q^{15} - 2022 q^{17} - 1732 q^{19} - 441 q^{21} + 576 q^{23} + 5711 q^{25} - 729 q^{27} + 5518 q^{29} - 6336 q^{31} + 468 q^{33} + 4606 q^{35} - 7338 q^{37} + 6930 q^{39} - 3262 q^{41} - 5420 q^{43} + 7614 q^{45} - 864 q^{47} + 2401 q^{49} + 18198 q^{51} + 4182 q^{53} - 4888 q^{55} + 15588 q^{57} + 11220 q^{59} - 45602 q^{61} + 3969 q^{63} - 72380 q^{65} - 1396 q^{67} - 5184 q^{69} - 18720 q^{71} + 46362 q^{73} - 51399 q^{75} - 2548 q^{77} - 97424 q^{79} + 6561 q^{81} + 81228 q^{83} - 190068 q^{85} - 49662 q^{87} - 3182 q^{89} - 37730 q^{91} + 57024 q^{93} - 162808 q^{95} + 4914 q^{97} - 4212 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

94.0000

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.i		1
3.b	odd	2	1		1008.6.a.a		1
4.b	odd	2	1		21.6.a.c	✓	1
12.b	even	2	1		63.6.a.b		1
20.d	odd	2	1		525.6.a.b		1
20.e	even	4	2		525.6.d.c		2
28.d	even	2	1		147.6.a.f		1
28.f	even	6	2		147.6.e.d		2
28.g	odd	6	2		147.6.e.c		2
84.h	odd	2	1		441.6.a.c		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.a.c	✓	1	4.b	odd	2	1
63.6.a.b		1	12.b	even	2	1
147.6.a.f		1	28.d	even	2	1
147.6.e.c		2	28.g	odd	6	2
147.6.e.d		2	28.f	even	6	2
336.6.a.i		1	1.a	even	1	1	trivial
441.6.a.c		1	84.h	odd	2	1
525.6.a.b		1	20.d	odd	2	1
525.6.d.c		2	20.e	even	4	2
1008.6.a.a		1	3.b	odd	2	1

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} - 94 $

$ T_{11} + 52 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 9 $ T + 9
$5$	$ T - 94 $ T - 94
$7$	$ T - 49 $ T - 49
$11$	$ T + 52 $ T + 52
$13$	$ T + 770 $ T + 770
$17$	$ T + 2022 $ T + 2022
$19$	$ T + 1732 $ T + 1732
$23$	$ T - 576 $ T - 576
$29$	$ T - 5518 $ T - 5518
$31$	$ T + 6336 $ T + 6336
$37$	$ T + 7338 $ T + 7338
$41$	$ T + 3262 $ T + 3262
$43$	$ T + 5420 $ T + 5420
$47$	$ T + 864 $ T + 864
$53$	$ T - 4182 $ T - 4182
$59$	$ T - 11220 $ T - 11220
$61$	$ T + 45602 $ T + 45602
$67$	$ T + 1396 $ T + 1396
$71$	$ T + 18720 $ T + 18720
$73$	$ T - 46362 $ T - 46362
$79$	$ T + 97424 $ T + 97424
$83$	$ T - 81228 $ T - 81228
$89$	$ T + 3182 $ T + 3182
$97$	$ T - 4914 $ T - 4914
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Elliptic curves over $\Q$
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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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