LMFDB - Newform orbit 5408.2.a.q

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, 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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5408 = 2^{5} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 416)
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 $\beta = \sqrt{5}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta q^{3} - 3 q^{5} + \beta q^{7} + 2 q^{9} - 2 \beta q^{11} + 3 \beta q^{15} - 3 q^{17} - 2 \beta q^{19} - 5 q^{21} + 4 \beta q^{23} + 4 q^{25} + \beta q^{27} + 10 q^{29} + 10 q^{33} - 3 \beta q^{35} + \cdots - 4 \beta q^{99} +O(q^{100})$ q - b * q^3 - 3 * q^5 + b * q^7 + 2 * q^9 - 2b q^11 + 3b q^15 - 3 * q^17 - 2b q^19 - 5 * q^21 + 4b q^23 + 4 * q^25 + b * q^27 + 10 * q^29 + 10 * q^33 - 3b q^35 - 3 * q^37 - 3b q^43 - 6 * q^45 + b * q^47 - 2 * q^49 + 3b q^51 + 4 * q^53 + 6b q^55 + 10 * q^57 - 2b q^59 + 2b q^63 + 6b q^67 - 20 * q^69 + 3b q^71 - 14 * q^73 - 4b q^75 - 10 * q^77 + 4b q^79 - 11 * q^81 + 8b q^83 + 9 * q^85 - 10b q^87 + 10 * q^89 + 6b q^95 + 2 * q^97 - 4b q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 6 q^{5} + 4 q^{9} - 6 q^{17} - 10 q^{21} + 8 q^{25} + 20 q^{29} + 20 q^{33} - 6 q^{37} - 12 q^{45} - 4 q^{49} + 8 q^{53} + 20 q^{57} - 40 q^{69} - 28 q^{73} - 20 q^{77} - 22 q^{81} + 18 q^{85} + 20 q^{89}+ \cdots + 4 q^{97}+O(q^{100})$ 2 * q - 6 * q^5 + 4 * q^9 - 6 * q^17 - 10 * q^21 + 8 * q^25 + 20 * q^29 + 20 * q^33 - 6 * q^37 - 12 * q^45 - 4 * q^49 + 8 * q^53 + 20 * q^57 - 40 * q^69 - 28 * q^73 - 20 * q^77 - 22 * q^81 + 18 * q^85 + 20 * q^89 + 4 * 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

−2.23607

−3.00000

2.23607

2.00000

1.2

2.23607

−3.00000

−2.23607

2.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$13$	$+1$

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	5408.2.a.q		2
4.b	odd	2	1	inner	5408.2.a.q		2
13.b	even	2	1		416.2.a.d	✓	2
39.d	odd	2	1		3744.2.a.q		2
52.b	odd	2	1		416.2.a.d	✓	2
104.e	even	2	1		832.2.a.m		2
104.h	odd	2	1		832.2.a.m		2
156.h	even	2	1		3744.2.a.q		2
208.o	odd	4	2		3328.2.b.x		4
208.p	even	4	2		3328.2.b.x		4
312.b	odd	2	1		7488.2.a.cw		2
312.h	even	2	1		7488.2.a.cw		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.a.d	✓	2	13.b	even	2	1
416.2.a.d	✓	2	52.b	odd	2	1
832.2.a.m		2	104.e	even	2	1
832.2.a.m		2	104.h	odd	2	1
3328.2.b.x		4	208.o	odd	4	2
3328.2.b.x		4	208.p	even	4	2
3744.2.a.q		2	39.d	odd	2	1
3744.2.a.q		2	156.h	even	2	1
5408.2.a.q		2	1.a	even	1	1	trivial
5408.2.a.q		2	4.b	odd	2	1	inner
7488.2.a.cw		2	312.b	odd	2	1
7488.2.a.cw		2	312.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5408))$ :

T_{3}^{2} - 5

T_{5} + 3

T_{7}^{2} - 5

T_{37} + 3

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 5$ T^2 - 5
$5$	$(T + 3)^{2}$ (T + 3)^2
$7$	$T^{2} - 5$ T^2 - 5
$11$	$T^{2} - 20$ T^2 - 20
$13$	$T^{2}$ T^2
$17$	$(T + 3)^{2}$ (T + 3)^2
$19$	$T^{2} - 20$ T^2 - 20
$23$	$T^{2} - 80$ T^2 - 80
$29$	$(T - 10)^{2}$ (T - 10)^2
$31$	$T^{2}$ T^2
$37$	$(T + 3)^{2}$ (T + 3)^2
$41$	$T^{2}$ T^2
$43$	$T^{2} - 45$ T^2 - 45
$47$	$T^{2} - 5$ T^2 - 5
$53$	$(T - 4)^{2}$ (T - 4)^2
$59$	$T^{2} - 20$ T^2 - 20
$61$	$T^{2}$ T^2
$67$	$T^{2} - 180$ T^2 - 180
$71$	$T^{2} - 45$ T^2 - 45
$73$	$(T + 14)^{2}$ (T + 14)^2
$79$	$T^{2} - 80$ T^2 - 80
$83$	$T^{2} - 320$ T^2 - 320
$89$	$(T - 10)^{2}$ (T - 10)^2
$97$	$(T - 2)^{2}$ (T - 2)^2
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