LMFDB - Newform orbit 5000.2.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5000,2,Mod(1,5000)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5000.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$5000 = 2^{3} \cdot 5^{4}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5000.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,3,0,4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.9252010106$
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 200)
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.

$f(q)$	$=$	$q + ( - 2 \beta + 1) q^{3} + (\beta + 1) q^{7} + 2 q^{9} + (2 \beta - 2) q^{11} + ( - \beta + 5) q^{13} + (2 \beta + 2) q^{17} + (\beta + 6) q^{19} + ( - 3 \beta - 1) q^{21} + (4 \beta - 1) q^{23} + (2 \beta - 1) q^{27}+ \cdots + (4 \beta - 4) q^{99}+O(q^{100})$ q + (-2b + 1) q^3 + (b + 1) * q^7 + 2 * q^9 + (2b - 2) q^11 + (-b + 5) * q^13 + (2b + 2) q^17 + (b + 6) * q^19 + (-3b - 1) q^21 + (4b - 1) q^23 + (2b - 1) q^27 + (-b + 6) * q^29 + (-6b + 5) q^31 + (2b - 6) q^33 + (-2b - 5) q^37 + (-9b + 7) q^39 + (6b - 4) q^41 + (-b + 6) * q^43 + (-3b + 5) q^47 + (3b - 5) q^49 + (-6b - 2) q^51 + (8b - 5) q^53 + (-13b + 4) q^57 + (-b + 4) * q^59 + (-10b + 3) q^61 + (2b + 2) q^63 + (6b + 4) q^67 + (-2b - 9) q^69 + (-b - 7) * q^71 + 11 * q^73 + 2b q^77 + (-7b + 6) q^79 - 11 * q^81 + (4b - 11) q^83 + (-11b + 8) q^87 + 4 * q^89 + (3b + 4) q^91 + (-4b + 17) q^93 + (-15b + 9) q^97 + (4b - 4) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{7} + 4 q^{9} - 2 q^{11} + 9 q^{13} + 6 q^{17} + 13 q^{19} - 5 q^{21} + 2 q^{23} + 11 q^{29} + 4 q^{31} - 10 q^{33} - 12 q^{37} + 5 q^{39} - 2 q^{41} + 11 q^{43} + 7 q^{47} - 7 q^{49} - 10 q^{51}+ \cdots - 4 q^{99}+O(q^{100})$ 2 * q + 3 * q^7 + 4 * q^9 - 2 * q^11 + 9 * q^13 + 6 * q^17 + 13 * q^19 - 5 * q^21 + 2 * q^23 + 11 * q^29 + 4 * q^31 - 10 * q^33 - 12 * q^37 + 5 * q^39 - 2 * q^41 + 11 * q^43 + 7 * q^47 - 7 * q^49 - 10 * q^51 - 2 * q^53 - 5 * q^57 + 7 * q^59 - 4 * q^61 + 6 * q^63 + 14 * q^67 - 20 * q^69 - 15 * q^71 + 22 * q^73 + 2 * q^77 + 5 * q^79 - 22 * q^81 - 18 * q^83 + 5 * q^87 + 8 * q^89 + 11 * q^91 + 30 * q^93 + 3 * q^97 - 4 * 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

1.61803
−0.618034

−2.23607

2.61803

2.00000

1.2

2.23607

0.381966

2.00000

Atkin-Lehner signs

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

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	5000.2.a.c		2
4.b	odd	2	1		10000.2.a.g		2
5.b	even	2	1		5000.2.a.a		2
20.d	odd	2	1		10000.2.a.i		2
25.d	even	5	2		200.2.m.a	✓	4
25.e	even	10	2		1000.2.m.a		4
25.f	odd	20	4		1000.2.q.a		8
100.j	odd	10	2		400.2.u.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.2.m.a	✓	4	25.d	even	5	2
400.2.u.a		4	100.j	odd	10	2
1000.2.m.a		4	25.e	even	10	2
1000.2.q.a		8	25.f	odd	20	4
5000.2.a.a		2	5.b	even	2	1
5000.2.a.c		2	1.a	even	1	1	trivial
10000.2.a.g		2	4.b	odd	2	1
10000.2.a.i		2	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(5000))$ :

T_{3}^{2} - 5

T3^2 - 5

T_{7}^{2} - 3T_{7} + 1

T7^2 - 3*T7 + 1

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

T11^2 + 2*T11 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 5$ T^2 - 5
$5$	$T^{2}$ T^2
$7$	$T^{2} - 3T + 1$ T^2 - 3*T + 1
$11$	$T^{2} + 2T - 4$ T^2 + 2*T - 4
$13$	$T^{2} - 9T + 19$ T^2 - 9*T + 19
$17$	$T^{2} - 6T + 4$ T^2 - 6*T + 4
$19$	$T^{2} - 13T + 41$ T^2 - 13*T + 41
$23$	$T^{2} - 2T - 19$ T^2 - 2*T - 19
$29$	$T^{2} - 11T + 29$ T^2 - 11*T + 29
$31$	$T^{2} - 4T - 41$ T^2 - 4*T - 41
$37$	$T^{2} + 12T + 31$ T^2 + 12*T + 31
$41$	$T^{2} + 2T - 44$ T^2 + 2*T - 44
$43$	$T^{2} - 11T + 29$ T^2 - 11*T + 29
$47$	$T^{2} - 7T + 1$ T^2 - 7*T + 1
$53$	$T^{2} + 2T - 79$ T^2 + 2*T - 79
$59$	$T^{2} - 7T + 11$ T^2 - 7*T + 11
$61$	$T^{2} + 4T - 121$ T^2 + 4*T - 121
$67$	$T^{2} - 14T + 4$ T^2 - 14*T + 4
$71$	$T^{2} + 15T + 55$ T^2 + 15*T + 55
$73$	$(T - 11)^{2}$ (T - 11)^2
$79$	$T^{2} - 5T - 55$ T^2 - 5*T - 55
$83$	$T^{2} + 18T + 61$ T^2 + 18*T + 61
$89$	$(T - 4)^{2}$ (T - 4)^2
$97$	$T^{2} - 3T - 279$ T^2 - 3*T - 279
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