LMFDB - Newform orbit 8800.2.a.bg

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8800,2,Mod(1,8800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8800 = 2^{5} \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8800.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:	$70.2683537787$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 4x - 1$ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 1760)
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)

Coefficients of the $q$ -expansion are expressed in terms of a basis $1,\beta_1,\beta_2$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta_{2} - 1) q^{3} + ( - \beta_1 + 1) q^{7} + (2 \beta_{2} - \beta_1 + 2) q^{9} - q^{11} - 2 q^{13} + ( - \beta_{2} - 1) q^{17} + (2 \beta_{2} + \beta_1 + 3) q^{19} + ( - 3 \beta_{2} - 1) q^{21}+ \cdots + ( - 2 \beta_{2} + \beta_1 - 2) q^{99}+O(q^{100})$ q + (-b2 - 1) * q^3 + (-b1 + 1) * q^7 + (2b2 - b1 + 2) q^9 - q^11 - 2 * q^13 + (-b2 - 1) * q^17 + (2b2 + b1 + 3) q^19 + (-3b2 - 1) q^21 + 2 * q^23 + (-3b2 + 2b1 - 7) * q^27 + (b2 + 2b1 - 1) q^29 + (3b2 + 1) q^31 + (b2 + 1) * q^33 + (b1 - 5) * q^37 + (2b2 + 2) q^39 + (-2b2 + 2b1 + 2) * q^41 + (4b2 - 2b1) * q^43 + (-2b2 + 2b1 + 2) * q^47 + (-2b2 - b1 + 2) q^49 + (2b2 - b1 + 5) q^51 + (b1 - 5) * q^53 + (-3b2 + 2b1 - 11) * q^57 + (-4b2 + 2b1 - 2) * q^59 + (-b2 + 2b1 + 5) q^61 + (4b2 + 10) q^63 + (2b1 - 4) q^67 + (-2b2 - 2) q^69 + (b2 + 3) * q^71 + (-2b2 - 2b1 - 10) * q^73 + (b1 - 1) * q^77 + (2b2 - 10) q^79 + (8b2 + 13) q^81 + (2b2 + 2b1 - 2) * q^83 + (4b2 + b1 - 3) q^87 + (2b2 - 3b1 + 1) * q^89 + (2b1 - 2) q^91 + (-4b2 + 3b1 - 13) * q^93 + (2b2 - 4b1) * q^97 + (-2b2 + b1 - 2) q^99
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 2 q^{3} + 4 q^{7} + 5 q^{9} - 3 q^{11} - 6 q^{13} - 2 q^{17} + 6 q^{19} + 6 q^{23} - 20 q^{27} - 6 q^{29} + 2 q^{33} - 16 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 6 q^{47} + 9 q^{49} + 14 q^{51}+ \cdots - 5 q^{99}+O(q^{100})$ 3 * q - 2 * q^3 + 4 * q^7 + 5 * q^9 - 3 * q^11 - 6 * q^13 - 2 * q^17 + 6 * q^19 + 6 * q^23 - 20 * q^27 - 6 * q^29 + 2 * q^33 - 16 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 6 * q^47 + 9 * q^49 + 14 * q^51 - 16 * q^53 - 32 * q^57 - 4 * q^59 + 14 * q^61 + 26 * q^63 - 14 * q^67 - 4 * q^69 + 8 * q^71 - 26 * q^73 - 4 * q^77 - 32 * q^79 + 31 * q^81 - 10 * q^83 - 14 * q^87 + 4 * q^89 - 8 * q^91 - 38 * q^93 + 2 * q^97 - 5 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{3} - 4x - 1$ x^3 - 4*x - 1 :

$\beta_{1}$	$=$	$\nu^{2} + \nu - 3$ v^2 + v - 3
$\beta_{2}$	$=$	$\nu^{2} - \nu - 3$ v^2 - v - 3

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( -\beta_{2} + \beta_1 ) / 2$ (-b2 + b1) / 2
$\nu^{2}$	$=$	$( \beta_{2} + \beta _1 + 6 ) / 2$ (b2 + b1 + 6) / 2

Display $\beta_i$ in terms of $\nu^j$

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

−1.86081
2.11491
−0.254102

−3.32340

2.39821

8.04502

1.2

−0.357926

−2.58774

−2.87189

1.3

1.68133

4.18953

−0.173127

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$11$	$+1$

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	8800.2.a.bg		3
4.b	odd	2	1		8800.2.a.bj		3
5.b	even	2	1		1760.2.a.t	yes	3
20.d	odd	2	1		1760.2.a.q	✓	3
40.e	odd	2	1		3520.2.a.bx		3
40.f	even	2	1		3520.2.a.bu		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1760.2.a.q	✓	3	20.d	odd	2	1
1760.2.a.t	yes	3	5.b	even	2	1
3520.2.a.bu		3	40.f	even	2	1
3520.2.a.bx		3	40.e	odd	2	1
8800.2.a.bg		3	1.a	even	1	1	trivial
8800.2.a.bj		3	4.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 $S_{2}^{\mathrm{new}}(\Gamma_0(8800))$ :

T_{3}^{3} + 2T_{3}^{2} - 5T_{3} - 2

T_{7}^{3} - 4T_{7}^{2} - 7T_{7} + 26

T_{13} + 2

T_{17}^{3} + 2T_{17}^{2} - 5T_{17} - 2

T_{19}^{3} - 6T_{19}^{2} - 31T_{19} + 184

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3} + 2 T^{2} + \cdots - 2$ T^3 + 2T^2 - 5T - 2
$5$	$T^{3}$ T^3
$7$	$T^{3} - 4 T^{2} + \cdots + 26$ T^3 - 4T^2 - 7T + 26
$11$	$(T + 1)^{3}$ (T + 1)^3
$13$	$(T + 2)^{3}$ (T + 2)^3
$17$	$T^{3} + 2 T^{2} + \cdots - 2$ T^3 + 2T^2 - 5T - 2
$19$	$T^{3} - 6 T^{2} + \cdots + 184$ T^3 - 6T^2 - 31T + 184
$23$	$(T - 2)^{3}$ (T - 2)^3
$29$	$T^{3} + 6 T^{2} + \cdots - 82$ T^3 + 6T^2 - 49T - 82
$31$	$T^{3} - 57T - 52$ T^3 - 57*T - 52
$37$	$T^{3} + 16 T^{2} + \cdots + 74$ T^3 + 16T^2 + 73T + 74
$41$	$T^{3} - 6 T^{2} + \cdots + 56$ T^3 - 6T^2 - 52T + 56
$43$	$T^{3} + 2 T^{2} + \cdots - 512$ T^3 + 2T^2 - 128T - 512
$47$	$T^{3} - 6 T^{2} + \cdots + 56$ T^3 - 6T^2 - 52T + 56
$53$	$T^{3} + 16 T^{2} + \cdots + 74$ T^3 + 16T^2 + 73T + 74
$59$	$T^{3} + 4 T^{2} + \cdots + 256$ T^3 + 4T^2 - 124T + 256
$61$	$T^{3} - 14 T^{2} + \cdots + 2$ T^3 - 14T^2 + 15T + 2
$67$	$T^{3} + 14 T^{2} + \cdots - 224$ T^3 + 14T^2 + 16T - 224
$71$	$T^{3} - 8 T^{2} + \cdots - 4$ T^3 - 8T^2 + 15T - 4
$73$	$T^{3} + 26 T^{2} + \cdots - 328$ T^3 + 26T^2 + 140T - 328
$79$	$T^{3} + 32 T^{2} + \cdots + 928$ T^3 + 32T^2 + 316T + 928
$83$	$T^{3} + 10 T^{2} + \cdots - 8$ T^3 + 10T^2 - 52T - 8
$89$	$T^{3} - 4 T^{2} + \cdots + 566$ T^3 - 4T^2 - 115T + 566
$97$	$T^{3} - 2 T^{2} + \cdots + 1184$ T^3 - 2T^2 - 200T + 1184
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