LMFDB - Newform orbit 6084.2.a.z

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6084,2,Mod(1,6084)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6084, 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("6084.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6084 = 2^{2} \cdot 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6084.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:	$48.5809845897$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 2x + 1$ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2028)
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.

$f(q)$	$=$	$q + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{2} - \beta_1 - 2) q^{7} + (\beta_1 + 2) q^{11} + ( - \beta_{2} + 2 \beta_1 - 2) q^{17} + ( - 3 \beta_{2} + 2 \beta_1 - 6) q^{19} + (3 \beta_{2} - \beta_1 + 6) q^{23}+ \cdots + ( - 2 \beta_{2} + 6 \beta_1 - 9) q^{97}+O(q^{100})$ q + (-b2 - b1) * q^5 + (-b2 - b1 - 2) * q^7 + (b1 + 2) * q^11 + (-b2 + 2b1 - 2) q^17 + (-3b2 + 2b1 - 6) * q^19 + (3b2 - b1 + 6) q^23 + (2b2 + b1) q^25 + (4b2 - 3b1 - 1) * q^29 + (4b2 + b1 - 1) q^31 + (4b2 + 3b1 + 5) * q^35 + (5b2 - 4b1 + 1) * q^37 + (4b2 - b1 + 6) q^41 + (4b2 - 4b1 + 1) * q^43 + (-b2 + 6b1) q^47 + (6b2 + 5b1 + 2) * q^49 + (-6b2 + 9b1 - 4) * q^53 + (-4b2 - 2b1 - 3) * q^55 + (2b2 - 10b1 + 2) * q^59 + (-5b2 + 6b1 - 4) * q^61 + (b2 + b1 - 11) * q^67 + (6b2 - 3b1 + 1) * q^71 + (-5b2 - b1 - 6) q^73 + (-4b2 - 4b1 - 7) * q^77 + (-b2 + 3b1 - 5) q^79 + (-9b2 + 8b1 - 3) * q^83 + (-2b2 + 3b1 - 4) * q^85 + (-9b2 + 5b1 - 7) * q^89 + (2b2 + 9b1) * q^95 + (-2b2 + 6b1 - 9) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 6 q^{7} + 7 q^{11} - 3 q^{17} - 13 q^{19} + 14 q^{23} - q^{25} - 10 q^{29} - 6 q^{31} + 14 q^{35} - 6 q^{37} + 13 q^{41} - 5 q^{43} + 7 q^{47} + 5 q^{49} + 3 q^{53} - 7 q^{55} - 6 q^{59} - q^{61}+ \cdots - 19 q^{97}+O(q^{100})$ 3 * q - 6 * q^7 + 7 * q^11 - 3 * q^17 - 13 * q^19 + 14 * q^23 - q^25 - 10 * q^29 - 6 * q^31 + 14 * q^35 - 6 * q^37 + 13 * q^41 - 5 * q^43 + 7 * q^47 + 5 * q^49 + 3 * q^53 - 7 * q^55 - 6 * q^59 - q^61 - 33 * q^67 - 6 * q^71 - 14 * q^73 - 21 * q^77 - 11 * q^79 + 8 * q^83 - 7 * q^85 - 7 * q^89 + 7 * q^95 - 19 * q^97

Basis of coefficient ring in terms of $\nu = \zeta_{14} + \zeta_{14}^{-1}$ :

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

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

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

1.80194
0.445042
−1.24698

−3.04892

−5.04892

1.2

1.35690

−0.643104

1.3

1.69202

−0.307979

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$13$	$+1$

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	6084.2.a.z		3
3.b	odd	2	1		2028.2.a.k	✓	3
12.b	even	2	1		8112.2.a.cd		3
13.b	even	2	1		6084.2.a.ba		3
13.d	odd	4	2		6084.2.b.q		6
39.d	odd	2	1		2028.2.a.l	yes	3
39.f	even	4	2		2028.2.b.g		6
39.h	odd	6	2		2028.2.i.j		6
39.i	odd	6	2		2028.2.i.k		6
39.k	even	12	4		2028.2.q.i		12
156.h	even	2	1		8112.2.a.ca		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2028.2.a.k	✓	3	3.b	odd	2	1
2028.2.a.l	yes	3	39.d	odd	2	1
2028.2.b.g		6	39.f	even	4	2
2028.2.i.j		6	39.h	odd	6	2
2028.2.i.k		6	39.i	odd	6	2
2028.2.q.i		12	39.k	even	12	4
6084.2.a.z		3	1.a	even	1	1	trivial
6084.2.a.ba		3	13.b	even	2	1
6084.2.b.q		6	13.d	odd	4	2
8112.2.a.ca		3	156.h	even	2	1
8112.2.a.cd		3	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6084))$ :

T_{5}^{3} - 7T_{5} + 7

T_{7}^{3} + 6T_{7}^{2} + 5T_{7} + 1

T_{11}^{3} - 7T_{11}^{2} + 14T_{11} - 7

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{3}$ T^3
$3$	$T^{3}$ T^3
$5$	$T^{3} - 7T + 7$ T^3 - 7*T + 7
$7$	$T^{3} + 6 T^{2} + \cdots + 1$ T^3 + 6T^2 + 5T + 1
$11$	$T^{3} - 7 T^{2} + \cdots - 7$ T^3 - 7T^2 + 14T - 7
$13$	$T^{3}$ T^3
$17$	$T^{3} + 3 T^{2} + \cdots + 1$ T^3 + 3T^2 - 4T + 1
$19$	$T^{3} + 13 T^{2} + \cdots - 13$ T^3 + 13T^2 + 40T - 13
$23$	$T^{3} - 14 T^{2} + \cdots - 7$ T^3 - 14T^2 + 49T - 7
$29$	$T^{3} + 10 T^{2} + \cdots - 13$ T^3 + 10T^2 + 3T - 13
$31$	$T^{3} + 6 T^{2} + \cdots - 181$ T^3 + 6T^2 - 37T - 181
$37$	$T^{3} + 6 T^{2} + \cdots + 1$ T^3 + 6T^2 - 37T + 1
$41$	$T^{3} - 13 T^{2} + \cdots + 83$ T^3 - 13T^2 + 26T + 83
$43$	$T^{3} + 5 T^{2} + \cdots - 41$ T^3 + 5T^2 - 29T - 41
$47$	$T^{3} - 7 T^{2} + \cdots + 301$ T^3 - 7T^2 - 56T + 301
$53$	$T^{3} - 3 T^{2} + \cdots + 643$ T^3 - 3T^2 - 144T + 643
$59$	$T^{3} + 6 T^{2} + \cdots - 1112$ T^3 + 6T^2 - 184T - 1112
$61$	$T^{3} + T^{2} + \cdots + 41$ T^3 + T^2 - 72*T + 41
$67$	$T^{3} + 33 T^{2} + \cdots + 1247$ T^3 + 33T^2 + 356T + 1247
$71$	$T^{3} + 6 T^{2} + \cdots + 71$ T^3 + 6T^2 - 51T + 71
$73$	$T^{3} + 14 T^{2} + \cdots - 91$ T^3 + 14T^2 - 7T - 91
$79$	$T^{3} + 11 T^{2} + \cdots + 13$ T^3 + 11T^2 + 24T + 13
$83$	$T^{3} - 8 T^{2} + \cdots + 29$ T^3 - 8T^2 - 149T + 29
$89$	$T^{3} + 7 T^{2} + \cdots - 973$ T^3 + 7T^2 - 126T - 973
$97$	$T^{3} + 19 T^{2} + \cdots + 29$ T^3 + 19T^2 + 55T + 29
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