LMFDB - Newform orbit 336.4.a.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,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, 4, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
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:	$19.8246417619$
Analytic rank:	$0$
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 + 3 q^{3} - 18 q^{5} - 7 q^{7} + 9 q^{9} + 36 q^{11} - 34 q^{13} - 54 q^{15} + 42 q^{17} + 124 q^{19} - 21 q^{21} + 199 q^{25} + 27 q^{27} + 102 q^{29} + 160 q^{31} + 108 q^{33} + 126 q^{35} + 398 q^{37}+ \cdots + 324 q^{99}+O(q^{100})

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

3.00000

−18.0000

−7.00000

9.00000

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	336.4.a.f		1
3.b	odd	2	1		1008.4.a.v		1
4.b	odd	2	1		21.4.a.a	✓	1
7.b	odd	2	1		2352.4.a.r		1
8.b	even	2	1		1344.4.a.n		1
8.d	odd	2	1		1344.4.a.ba		1
12.b	even	2	1		63.4.a.c		1
20.d	odd	2	1		525.4.a.g		1
20.e	even	4	2		525.4.d.c		2
28.d	even	2	1		147.4.a.c		1
28.f	even	6	2		147.4.e.g		2
28.g	odd	6	2		147.4.e.i		2
60.h	even	2	1		1575.4.a.b		1
84.h	odd	2	1		441.4.a.j		1
84.j	odd	6	2		441.4.e.d		2
84.n	even	6	2		441.4.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.4.a.a	✓	1	4.b	odd	2	1
63.4.a.c		1	12.b	even	2	1
147.4.a.c		1	28.d	even	2	1
147.4.e.g		2	28.f	even	6	2
147.4.e.i		2	28.g	odd	6	2
336.4.a.f		1	1.a	even	1	1	trivial
441.4.a.j		1	84.h	odd	2	1
441.4.e.b		2	84.n	even	6	2
441.4.e.d		2	84.j	odd	6	2
525.4.a.g		1	20.d	odd	2	1
525.4.d.c		2	20.e	even	4	2
1008.4.a.v		1	3.b	odd	2	1
1344.4.a.n		1	8.b	even	2	1
1344.4.a.ba		1	8.d	odd	2	1
1575.4.a.b		1	60.h	even	2	1
2352.4.a.r		1	7.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 $S_{4}^{\mathrm{new}}(\Gamma_0(336))$ :

T_{5} + 18

T_{11} - 36

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 3$ T - 3
$5$	$T + 18$ T + 18
$7$	$T + 7$ T + 7
$11$	$T - 36$ T - 36
$13$	$T + 34$ T + 34
$17$	$T - 42$ T - 42
$19$	$T - 124$ T - 124
$23$	$T$ T
$29$	$T - 102$ T - 102
$31$	$T - 160$ T - 160
$37$	$T - 398$ T - 398
$41$	$T + 318$ T + 318
$43$	$T - 268$ T - 268
$47$	$T + 240$ T + 240
$53$	$T + 498$ T + 498
$59$	$T - 132$ T - 132
$61$	$T - 398$ T - 398
$67$	$T + 92$ T + 92
$71$	$T - 720$ T - 720
$73$	$T + 502$ T + 502
$79$	$T - 1024$ T - 1024
$83$	$T - 204$ T - 204
$89$	$T - 354$ T - 354
$97$	$T + 286$ T + 286
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