LMFDB - Newform orbit 176.6.a.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,6,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("176.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$176 = 2^{4} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	176.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:	$28.2275522871$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 22)
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 - q^{3} - 51 q^{5} + 166 q^{7} - 242 q^{9} + 121 q^{11} + 692 q^{13} + 51 q^{15} - 738 q^{17} - 1424 q^{19} - 166 q^{21} + 1779 q^{23} - 524 q^{25} + 485 q^{27} - 2064 q^{29} - 6245 q^{31} - 121 q^{33}+ \cdots - 29282 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

−1.00000

−51.0000

166.000

−242.000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$11$	$-1$

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	176.6.a.b		1
4.b	odd	2	1		22.6.a.b	✓	1
8.b	even	2	1		704.6.a.f		1
8.d	odd	2	1		704.6.a.e		1
12.b	even	2	1		198.6.a.i		1
20.d	odd	2	1		550.6.a.f		1
20.e	even	4	2		550.6.b.f		2
28.d	even	2	1		1078.6.a.a		1
44.c	even	2	1		242.6.a.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
22.6.a.b	✓	1	4.b	odd	2	1
176.6.a.b		1	1.a	even	1	1	trivial
198.6.a.i		1	12.b	even	2	1
242.6.a.d		1	44.c	even	2	1
550.6.a.f		1	20.d	odd	2	1
550.6.b.f		2	20.e	even	4	2
704.6.a.e		1	8.d	odd	2	1
704.6.a.f		1	8.b	even	2	1
1078.6.a.a		1	28.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} + 1$ T3 + 1 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(176))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T + 1$ T + 1
$5$	$T + 51$ T + 51
$7$	$T - 166$ T - 166
$11$	$T - 121$ T - 121
$13$	$T - 692$ T - 692
$17$	$T + 738$ T + 738
$19$	$T + 1424$ T + 1424
$23$	$T - 1779$ T - 1779
$29$	$T + 2064$ T + 2064
$31$	$T + 6245$ T + 6245
$37$	$T + 14785$ T + 14785
$41$	$T - 5304$ T - 5304
$43$	$T + 17798$ T + 17798
$47$	$T - 17184$ T - 17184
$53$	$T + 30726$ T + 30726
$59$	$T - 34989$ T - 34989
$61$	$T + 45940$ T + 45940
$67$	$T + 25343$ T + 25343
$71$	$T + 13311$ T + 13311
$73$	$T + 53260$ T + 53260
$79$	$T + 77234$ T + 77234
$83$	$T + 55014$ T + 55014
$89$	$T - 125415$ T - 125415
$97$	$T + 88807$ T + 88807
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