LMFDB - Newform orbit 8512.2.a.k

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8512,2,Mod(1,8512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$8512 = 2^{6} \cdot 7 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-3,0,2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 532)
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.

$f(q)$	$=$	$q + ( - \beta - 1) q^{3} + q^{5} - q^{7} + (3 \beta - 1) q^{9} + (5 \beta - 2) q^{11} + (2 \beta - 1) q^{13} + ( - \beta - 1) q^{15} + ( - \beta - 6) q^{17} - q^{19} + (\beta + 1) q^{21} + (6 \beta - 1) q^{23}+ \cdots + (4 \beta + 17) q^{99}+O(q^{100})$ q + (-b - 1) * q^3 + q^5 - q^7 + (3b - 1) q^9 + (5b - 2) q^11 + (2b - 1) q^13 + (-b - 1) * q^15 + (-b - 6) * q^17 - q^19 + (b + 1) * q^21 + (6b - 1) q^23 - 4 * q^25 + (-2b + 1) q^27 + (-3b + 4) q^29 + (-b + 4) * q^31 + (-8b - 3) q^33 - q^35 + (6b - 3) q^37 + (-3b - 1) q^39 + (5b - 8) q^41 + (4b + 2) q^43 + (3b - 1) q^45 + (-6b + 5) q^47 + q^49 + (8b + 7) q^51 + (7b + 3) q^53 + (5b - 2) q^55 + (b + 1) * q^57 + (-6b - 1) q^59 + (-4b + 1) q^61 + (-3b + 1) q^63 + (2b - 1) q^65 + (-5b + 6) q^67 + (-11b - 5) q^69 + (10b - 5) q^71 + (5b - 9) q^73 + (4b + 4) q^75 + (-5b + 2) q^77 + (8b - 6) q^79 + (-6b + 4) q^81 + (b + 13) * q^83 + (-b - 6) * q^85 + (2b - 1) q^87 + (-10b + 4) q^89 + (-2b + 1) q^91 + (-2b - 3) q^93 - q^95 + (6b - 1) q^97 + (4b + 17) q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{3} + 2 q^{5} - 2 q^{7} + q^{9} + q^{11} - 3 q^{15} - 13 q^{17} - 2 q^{19} + 3 q^{21} + 4 q^{23} - 8 q^{25} + 5 q^{29} + 7 q^{31} - 14 q^{33} - 2 q^{35} - 5 q^{39} - 11 q^{41} + 8 q^{43} + q^{45}+ \cdots + 38 q^{99}+O(q^{100})$ 2 * q - 3 * q^3 + 2 * q^5 - 2 * q^7 + q^9 + q^11 - 3 * q^15 - 13 * q^17 - 2 * q^19 + 3 * q^21 + 4 * q^23 - 8 * q^25 + 5 * q^29 + 7 * q^31 - 14 * q^33 - 2 * q^35 - 5 * q^39 - 11 * q^41 + 8 * q^43 + q^45 + 4 * q^47 + 2 * q^49 + 22 * q^51 + 13 * q^53 + q^55 + 3 * q^57 - 8 * q^59 - 2 * q^61 - q^63 + 7 * q^67 - 21 * q^69 - 13 * q^73 + 12 * q^75 - q^77 - 4 * q^79 + 2 * q^81 + 27 * q^83 - 13 * q^85 - 2 * q^89 - 8 * q^93 - 2 * q^95 + 4 * q^97 + 38 * 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

−2.61803

1.00000

−1.00000

3.85410

1.2

−0.381966

1.00000

−1.00000

−2.85410

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$+1$
$19$	$+1$

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	8512.2.a.k		2
4.b	odd	2	1		8512.2.a.bg		2
8.b	even	2	1		2128.2.a.m		2
8.d	odd	2	1		532.2.a.b	✓	2
24.f	even	2	1		4788.2.a.l		2
56.e	even	2	1		3724.2.a.g		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
532.2.a.b	✓	2	8.d	odd	2	1
2128.2.a.m		2	8.b	even	2	1
3724.2.a.g		2	56.e	even	2	1
4788.2.a.l		2	24.f	even	2	1
8512.2.a.k		2	1.a	even	1	1	trivial
8512.2.a.bg		2	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8512))$ :

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

T3^2 + 3*T3 + 1

T_{5} - 1

T5 - 1

T_{11}^{2} - T_{11} - 31

T11^2 - T11 - 31

T_{23}^{2} - 4T_{23} - 41

T23^2 - 4*T23 - 41

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 3T + 1$ T^2 + 3*T + 1
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$(T + 1)^{2}$ (T + 1)^2
$11$	$T^{2} - T - 31$ T^2 - T - 31
$13$	$T^{2} - 5$ T^2 - 5
$17$	$T^{2} + 13T + 41$ T^2 + 13*T + 41
$19$	$(T + 1)^{2}$ (T + 1)^2
$23$	$T^{2} - 4T - 41$ T^2 - 4*T - 41
$29$	$T^{2} - 5T - 5$ T^2 - 5*T - 5
$31$	$T^{2} - 7T + 11$ T^2 - 7*T + 11
$37$	$T^{2} - 45$ T^2 - 45
$41$	$T^{2} + 11T - 1$ T^2 + 11*T - 1
$43$	$T^{2} - 8T - 4$ T^2 - 8*T - 4
$47$	$T^{2} - 4T - 41$ T^2 - 4*T - 41
$53$	$T^{2} - 13T - 19$ T^2 - 13*T - 19
$59$	$T^{2} + 8T - 29$ T^2 + 8*T - 29
$61$	$T^{2} + 2T - 19$ T^2 + 2*T - 19
$67$	$T^{2} - 7T - 19$ T^2 - 7*T - 19
$71$	$T^{2} - 125$ T^2 - 125
$73$	$T^{2} + 13T + 11$ T^2 + 13*T + 11
$79$	$T^{2} + 4T - 76$ T^2 + 4*T - 76
$83$	$T^{2} - 27T + 181$ T^2 - 27*T + 181
$89$	$T^{2} + 2T - 124$ T^2 + 2*T - 124
$97$	$T^{2} - 4T - 41$ T^2 - 4*T - 41
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