LMFDB - Newform orbit 5712.2.a.be

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5712,2,Mod(1,5712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5712, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5712.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5712 = 2^{4} \cdot 3 \cdot 7 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5712.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:	$45.6105496346$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 4$ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2856)
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 $\beta = \frac{1}{2}(1 + \sqrt{17})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - q^{3} + ( - \beta - 1) q^{5} + q^{7} + q^{9} + ( - \beta - 3) q^{11} + (3 \beta - 1) q^{13} + (\beta + 1) q^{15} - q^{17} + (2 \beta + 2) q^{19} - q^{21} + (2 \beta + 2) q^{23} + 3 \beta q^{25}+ \cdots + ( - \beta - 3) q^{99}+O(q^{100})$ q - q^3 + (-b - 1) * q^5 + q^7 + q^9 + (-b - 3) * q^11 + (3b - 1) q^13 + (b + 1) * q^15 - q^17 + (2b + 2) q^19 - q^21 + (2b + 2) q^23 + 3b q^25 - q^27 + (-4b + 2) q^29 - 8 * q^31 + (b + 3) * q^33 + (-b - 1) * q^35 + (-3b + 5) q^37 + (-3b + 1) q^39 + (-2b + 8) q^41 + (-b - 7) * q^43 + (-b - 1) * q^45 - 8 * q^47 + q^49 + q^51 + (b + 1) * q^53 + (5b + 7) q^55 + (-2b - 2) q^57 - 4 * q^59 - 2 * q^61 + q^63 + (-5b - 11) q^65 + (-3b + 3) q^67 + (-2b - 2) q^69 + (-4b + 8) q^71 + (-3b + 9) q^73 - 3b q^75 + (-b - 3) * q^77 + (b + 3) * q^79 + q^81 + (-3b - 5) q^83 + (b + 1) * q^85 + (4b - 2) q^87 + (b + 9) * q^89 + (3b - 1) q^91 + 8 * q^93 + (-6b - 10) q^95 + (b + 13) * q^97 + (-b - 3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 2 q^{9} - 7 q^{11} + q^{13} + 3 q^{15} - 2 q^{17} + 6 q^{19} - 2 q^{21} + 6 q^{23} + 3 q^{25} - 2 q^{27} - 16 q^{31} + 7 q^{33} - 3 q^{35} + 7 q^{37} - q^{39} + 14 q^{41}+ \cdots - 7 q^{99}+O(q^{100})$ 2 * q - 2 * q^3 - 3 * q^5 + 2 * q^7 + 2 * q^9 - 7 * q^11 + q^13 + 3 * q^15 - 2 * q^17 + 6 * q^19 - 2 * q^21 + 6 * q^23 + 3 * q^25 - 2 * q^27 - 16 * q^31 + 7 * q^33 - 3 * q^35 + 7 * q^37 - q^39 + 14 * q^41 - 15 * q^43 - 3 * q^45 - 16 * q^47 + 2 * q^49 + 2 * q^51 + 3 * q^53 + 19 * q^55 - 6 * q^57 - 8 * q^59 - 4 * q^61 + 2 * q^63 - 27 * q^65 + 3 * q^67 - 6 * q^69 + 12 * q^71 + 15 * q^73 - 3 * q^75 - 7 * q^77 + 7 * q^79 + 2 * q^81 - 13 * q^83 + 3 * q^85 + 19 * q^89 + q^91 + 16 * q^93 - 26 * q^95 + 27 * q^97 - 7 * q^99

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

2.56155
−1.56155

−1.00000

−3.56155

1.00000

1.2

−1.00000

0.561553

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$+1$
$7$	$-1$
$17$	$+1$

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	5712.2.a.be		2
4.b	odd	2	1		2856.2.a.m	✓	2
12.b	even	2	1		8568.2.a.r		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2856.2.a.m	✓	2	4.b	odd	2	1
5712.2.a.be		2	1.a	even	1	1	trivial
8568.2.a.r		2	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 $S_{2}^{\mathrm{new}}(\Gamma_0(5712))$ :

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

T_{11}^{2} + 7T_{11} + 8

T_{13}^{2} - T_{13} - 38

T_{19}^{2} - 6T_{19} - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T + 1)^{2}$ (T + 1)^2
$5$	$T^{2} + 3T - 2$ T^2 + 3*T - 2
$7$	$(T - 1)^{2}$ (T - 1)^2
$11$	$T^{2} + 7T + 8$ T^2 + 7*T + 8
$13$	$T^{2} - T - 38$ T^2 - T - 38
$17$	$(T + 1)^{2}$ (T + 1)^2
$19$	$T^{2} - 6T - 8$ T^2 - 6*T - 8
$23$	$T^{2} - 6T - 8$ T^2 - 6*T - 8
$29$	$T^{2} - 68$ T^2 - 68
$31$	$(T + 8)^{2}$ (T + 8)^2
$37$	$T^{2} - 7T - 26$ T^2 - 7*T - 26
$41$	$T^{2} - 14T + 32$ T^2 - 14*T + 32
$43$	$T^{2} + 15T + 52$ T^2 + 15*T + 52
$47$	$(T + 8)^{2}$ (T + 8)^2
$53$	$T^{2} - 3T - 2$ T^2 - 3*T - 2
$59$	$(T + 4)^{2}$ (T + 4)^2
$61$	$(T + 2)^{2}$ (T + 2)^2
$67$	$T^{2} - 3T - 36$ T^2 - 3*T - 36
$71$	$T^{2} - 12T - 32$ T^2 - 12*T - 32
$73$	$T^{2} - 15T + 18$ T^2 - 15*T + 18
$79$	$T^{2} - 7T + 8$ T^2 - 7*T + 8
$83$	$T^{2} + 13T + 4$ T^2 + 13*T + 4
$89$	$T^{2} - 19T + 86$ T^2 - 19*T + 86
$97$	$T^{2} - 27T + 178$ T^2 - 27*T + 178
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