LMFDB - Newform orbit 8000.2.a.bh

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8000,2,Mod(1,8000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8000 = 2^{6} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8000.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:	$63.8803216170$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - 5x^{2} + 5$ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 4000)
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} - \beta_{3} q^{7} + \beta_{2} q^{9} - 2 \beta_{3} q^{11} + (2 \beta_{2} + 2) q^{13} - 2 q^{17} + (2 \beta_{3} - 2 \beta_1) q^{19} + ( - 2 \beta_{2} - 1) q^{21} + \beta_{3} q^{23} + (\beta_{3} - 3 \beta_1) q^{27}+ \cdots + (2 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^3 - b3 * q^7 + b2 * q^9 - 2b3 q^11 + (2b2 + 2) q^13 - 2 * q^17 + (2b3 - 2b1) * q^19 + (-2b2 - 1) q^21 + b3 * q^23 + (b3 - 3b1) q^27 + (b2 - 2) * q^29 + (2b3 + 4b1) * q^31 + (-4b2 - 2) q^33 - 6b2 q^37 + (2b3 + 2b1) * q^39 + (-3b2 - 4) q^41 + (4b3 - 3b1) * q^43 + (2b3 - 5b1) * q^47 + (-b2 - 5) * q^49 - 2b1 q^51 + 4 * q^53 + (2b2 - 4) q^57 + 2b3 q^59 + (-b2 - 3) * q^61 + (b3 - b1) * q^63 + (-2b3 - 2b1) * q^67 + (2b2 + 1) q^69 + (6b3 + 2b1) * q^71 + (-2b2 - 10) q^73 + (-2b2 + 4) q^77 + (-8b3 - 2b1) * q^79 + (-4b2 - 8) q^81 + (-b3 + 2b1) q^83 + (b3 - 2b1) q^87 + (5b2 + 5) q^89 - 2b1 q^91 + (8b2 + 14) q^93 + (10b2 + 2) q^97 + (2b3 - 2b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{9} + 4 q^{13} - 8 q^{17} - 10 q^{29} + 12 q^{37} - 10 q^{41} - 18 q^{49} + 16 q^{53} - 20 q^{57} - 10 q^{61} - 36 q^{73} + 20 q^{77} - 24 q^{81} + 10 q^{89} + 40 q^{93} - 12 q^{97}+O(q^{100})$ 4 * q - 2 * q^9 + 4 * q^13 - 8 * q^17 - 10 * q^29 + 12 * q^37 - 10 * q^41 - 18 * q^49 + 16 * q^53 - 20 * q^57 - 10 * q^61 - 36 * q^73 + 20 * q^77 - 24 * q^81 + 10 * q^89 + 40 * q^93 - 12 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{20} + \zeta_{20}^{-1}$ :

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 3$ b2 + 3
$\nu^{3}$	$=$	$\beta_{3} + 3\beta_1$ b3 + 3*b1

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

−1.90211
−1.17557
1.17557
1.90211

−1.90211

1.17557

0.618034

1.2

−1.17557

−1.90211

−1.61803

1.3

1.17557

1.90211

−1.61803

1.4

1.90211

−1.17557

0.618034

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8000.2.a.bh		4
4.b	odd	2	1	inner	8000.2.a.bh		4
5.b	even	2	1		8000.2.a.bg		4
8.b	even	2	1		4000.2.a.e	✓	4
8.d	odd	2	1		4000.2.a.e	✓	4
20.d	odd	2	1		8000.2.a.bg		4
40.e	odd	2	1		4000.2.a.f	yes	4
40.f	even	2	1		4000.2.a.f	yes	4
40.i	odd	4	2		4000.2.c.d		8
40.k	even	4	2		4000.2.c.d		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.e	✓	4	8.b	even	2	1
4000.2.a.e	✓	4	8.d	odd	2	1
4000.2.a.f	yes	4	40.e	odd	2	1
4000.2.a.f	yes	4	40.f	even	2	1
4000.2.c.d		8	40.i	odd	4	2
4000.2.c.d		8	40.k	even	4	2
8000.2.a.bg		4	5.b	even	2	1
8000.2.a.bg		4	20.d	odd	2	1
8000.2.a.bh		4	1.a	even	1	1	trivial
8000.2.a.bh		4	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8000))$ :

T_{3}^{4} - 5T_{3}^{2} + 5

T_{7}^{4} - 5T_{7}^{2} + 5

T_{11}^{4} - 20T_{11}^{2} + 80

T_{13}^{2} - 2T_{13} - 4

T_{19}^{4} - 40T_{19}^{2} + 80

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} - 5T^{2} + 5$ T^4 - 5*T^2 + 5
$5$	$T^{4}$ T^4
$7$	$T^{4} - 5T^{2} + 5$ T^4 - 5*T^2 + 5
$11$	$T^{4} - 20T^{2} + 80$ T^4 - 20*T^2 + 80
$13$	$(T^{2} - 2 T - 4)^{2}$ (T^2 - 2*T - 4)^2
$17$	$(T + 2)^{4}$ (T + 2)^4
$19$	$T^{4} - 40T^{2} + 80$ T^4 - 40*T^2 + 80
$23$	$T^{4} - 5T^{2} + 5$ T^4 - 5*T^2 + 5
$29$	$(T^{2} + 5 T + 5)^{2}$ (T^2 + 5*T + 5)^2
$31$	$T^{4} - 100T^{2} + 80$ T^4 - 100*T^2 + 80
$37$	$(T^{2} - 6 T - 36)^{2}$ (T^2 - 6*T - 36)^2
$41$	$(T^{2} + 5 T - 5)^{2}$ (T^2 + 5*T - 5)^2
$43$	$T^{4} - 125T^{2} + 125$ T^4 - 125*T^2 + 125
$47$	$T^{4} - 145T^{2} + 4805$ T^4 - 145*T^2 + 4805
$53$	$(T - 4)^{4}$ (T - 4)^4
$59$	$T^{4} - 20T^{2} + 80$ T^4 - 20*T^2 + 80
$61$	$(T^{2} + 5 T + 5)^{2}$ (T^2 + 5*T + 5)^2
$67$	$T^{4} - 40T^{2} + 80$ T^4 - 40*T^2 + 80
$71$	$T^{4} - 200T^{2} + 9680$ T^4 - 200*T^2 + 9680
$73$	$(T^{2} + 18 T + 76)^{2}$ (T^2 + 18*T + 76)^2
$79$	$T^{4} - 340 T^{2} + 28880$ T^4 - 340*T^2 + 28880
$83$	$T^{4} - 25T^{2} + 125$ T^4 - 25*T^2 + 125
$89$	$(T^{2} - 5 T - 25)^{2}$ (T^2 - 5*T - 25)^2
$97$	$(T^{2} + 6 T - 116)^{2}$ (T^2 + 6*T - 116)^2
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