LMFDB - Newform orbit 693.4.a.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$693 = 3^{2} \cdot 7 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 231)
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^{2} + q^{4} + 14 q^{5} - 7 q^{7} + 21 q^{8} - 42 q^{10} + 11 q^{11} + 2 q^{13} + 21 q^{14} - 71 q^{16} + 74 q^{17} + 14 q^{20} - 33 q^{22} + 148 q^{23} + 71 q^{25} - 6 q^{26} - 7 q^{28} - 26 q^{29}+ \cdots - 147 q^{98}+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

1.00000

14.0000

−7.00000

21.0000

−42.0000

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$7$	$+1$
$11$	$-1$

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	693.4.a.c		1
3.b	odd	2	1		231.4.a.d	✓	1
21.c	even	2	1		1617.4.a.f		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.4.a.d	✓	1	3.b	odd	2	1
693.4.a.c		1	1.a	even	1	1	trivial
1617.4.a.f		1	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(693))$ :

T_{2} + 3

T_{5} - 14

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 3$ T + 3
$3$	$T$ T
$5$	$T - 14$ T - 14
$7$	$T + 7$ T + 7
$11$	$T - 11$ T - 11
$13$	$T - 2$ T - 2
$17$	$T - 74$ T - 74
$19$	$T$ T
$23$	$T - 148$ T - 148
$29$	$T + 26$ T + 26
$31$	$T - 112$ T - 112
$37$	$T + 98$ T + 98
$41$	$T - 10$ T - 10
$43$	$T - 208$ T - 208
$47$	$T + 460$ T + 460
$53$	$T + 258$ T + 258
$59$	$T - 204$ T - 204
$61$	$T - 178$ T - 178
$67$	$T + 924$ T + 924
$71$	$T - 748$ T - 748
$73$	$T + 230$ T + 230
$79$	$T + 456$ T + 456
$83$	$T - 228$ T - 228
$89$	$T - 198$ T - 198
$97$	$T - 562$ T - 562
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