LMFDB - Newform orbit 9000.2.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9000,2,Mod(1,9000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9000, 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("9000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9000 = 2^{3} \cdot 3^{2} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9000.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:	$71.8653618192$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 3000)
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 $\beta = \frac{1}{2}(1 + \sqrt{5})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + ( - \beta + 3) q^{7} + q^{11} + (2 \beta - 3) q^{13} + (3 \beta - 1) q^{17} + ( - 2 \beta - 3) q^{19} + ( - 6 \beta + 1) q^{23} + ( - 2 \beta + 5) q^{29} + (5 \beta - 7) q^{31} + (2 \beta - 5) q^{37}+ \cdots + (11 \beta - 1) q^{97}+O(q^{100})$ q + (-b + 3) * q^7 + q^11 + (2b - 3) q^13 + (3b - 1) q^17 + (-2b - 3) q^19 + (-6b + 1) q^23 + (-2b + 5) q^29 + (5b - 7) q^31 + (2b - 5) q^37 + (5b + 1) q^41 + (-3b + 8) q^43 + (2b - 1) q^47 + (-5b + 3) q^49 + (b + 4) * q^53 + (3b + 7) q^59 + (3b - 1) q^61 + (-4b + 8) q^67 + (-9b + 4) q^71 + (-3b + 4) q^73 + (-b + 3) * q^77 + (-6b + 10) q^79 + (-5b + 9) q^83 + 11 * q^89 + (7b - 11) q^91 + (11b - 1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 5 q^{7} + 2 q^{11} - 4 q^{13} + q^{17} - 8 q^{19} - 4 q^{23} + 8 q^{29} - 9 q^{31} - 8 q^{37} + 7 q^{41} + 13 q^{43} + q^{49} + 9 q^{53} + 17 q^{59} + q^{61} + 12 q^{67} - q^{71} + 5 q^{73} + 5 q^{77}+ \cdots + 9 q^{97}+O(q^{100})$ 2 * q + 5 * q^7 + 2 * q^11 - 4 * q^13 + q^17 - 8 * q^19 - 4 * q^23 + 8 * q^29 - 9 * q^31 - 8 * q^37 + 7 * q^41 + 13 * q^43 + q^49 + 9 * q^53 + 17 * q^59 + q^61 + 12 * q^67 - q^71 + 5 * q^73 + 5 * q^77 + 14 * q^79 + 13 * q^83 + 22 * q^89 - 15 * q^91 + 9 * q^97

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

1.61803
−0.618034

1.38197

1.2

3.61803

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$5$	$+1$

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	9000.2.a.p		2
3.b	odd	2	1		3000.2.a.d	✓	2
5.b	even	2	1		9000.2.a.a		2
12.b	even	2	1		6000.2.a.p		2
15.d	odd	2	1		3000.2.a.e	yes	2
15.e	even	4	2		3000.2.f.c		4
60.h	even	2	1		6000.2.a.l		2
60.l	odd	4	2		6000.2.f.i		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3000.2.a.d	✓	2	3.b	odd	2	1
3000.2.a.e	yes	2	15.d	odd	2	1
3000.2.f.c		4	15.e	even	4	2
6000.2.a.l		2	60.h	even	2	1
6000.2.a.p		2	12.b	even	2	1
6000.2.f.i		4	60.l	odd	4	2
9000.2.a.a		2	5.b	even	2	1
9000.2.a.p		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9000))$ :

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

T_{11} - 1

T_{13}^{2} + 4T_{13} - 1

T_{17}^{2} - T_{17} - 11

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2} - 5T + 5$ T^2 - 5*T + 5
$11$	$(T - 1)^{2}$ (T - 1)^2
$13$	$T^{2} + 4T - 1$ T^2 + 4*T - 1
$17$	$T^{2} - T - 11$ T^2 - T - 11
$19$	$T^{2} + 8T + 11$ T^2 + 8*T + 11
$23$	$T^{2} + 4T - 41$ T^2 + 4*T - 41
$29$	$T^{2} - 8T + 11$ T^2 - 8*T + 11
$31$	$T^{2} + 9T - 11$ T^2 + 9*T - 11
$37$	$T^{2} + 8T + 11$ T^2 + 8*T + 11
$41$	$T^{2} - 7T - 19$ T^2 - 7*T - 19
$43$	$T^{2} - 13T + 31$ T^2 - 13*T + 31
$47$	$T^{2} - 5$ T^2 - 5
$53$	$T^{2} - 9T + 19$ T^2 - 9*T + 19
$59$	$T^{2} - 17T + 61$ T^2 - 17*T + 61
$61$	$T^{2} - T - 11$ T^2 - T - 11
$67$	$T^{2} - 12T + 16$ T^2 - 12*T + 16
$71$	$T^{2} + T - 101$ T^2 + T - 101
$73$	$T^{2} - 5T - 5$ T^2 - 5*T - 5
$79$	$T^{2} - 14T + 4$ T^2 - 14*T + 4
$83$	$T^{2} - 13T + 11$ T^2 - 13*T + 11
$89$	$(T - 11)^{2}$ (T - 11)^2
$97$	$T^{2} - 9T - 131$ T^2 - 9*T - 131
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