LMFDB - Newform orbit 1512.2.a.q

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1512,2,Mod(1,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, 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("1512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1512 = 2^{3} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,2,0,0,0,0] 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:	$12.0733807856$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 8$ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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 $\beta = \frac{1}{2}(1 + \sqrt{33})$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{5} + q^{7} + (\beta + 1) q^{13} + q^{17} + ( - 2 \beta + 2) q^{19} + (\beta - 3) q^{23} + (\beta + 3) q^{25} + (\beta + 3) q^{29} + (\beta + 5) q^{31} + \beta q^{35} + ( - \beta + 2) q^{37}+ \cdots - 4 \beta q^{97} +O(q^{100})$ q + b * q^5 + q^7 + (b + 1) * q^13 + q^17 + (-2b + 2) q^19 + (b - 3) * q^23 + (b + 3) * q^25 + (b + 3) * q^29 + (b + 5) * q^31 + b * q^35 + (-b + 2) * q^37 + (-3b - 2) q^41 + (-2b + 3) q^43 + (-b - 2) * q^47 + q^49 + (-3b + 5) q^53 + (-2b - 5) q^59 + (2b + 6) q^61 + (2b + 8) q^65 + (b + 7) * q^67 + (b - 5) * q^71 + 10 * q^73 + (3b - 2) q^79 + (-5b + 4) q^83 + b * q^85 + (b + 5) * q^89 + (b + 1) * q^91 - 16 * q^95 - 4b q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{5} + 2 q^{7} + 3 q^{13} + 2 q^{17} + 2 q^{19} - 5 q^{23} + 7 q^{25} + 7 q^{29} + 11 q^{31} + q^{35} + 3 q^{37} - 7 q^{41} + 4 q^{43} - 5 q^{47} + 2 q^{49} + 7 q^{53} - 12 q^{59} + 14 q^{61} + 18 q^{65}+ \cdots - 4 q^{97}+O(q^{100})$ 2 * q + q^5 + 2 * q^7 + 3 * q^13 + 2 * q^17 + 2 * q^19 - 5 * q^23 + 7 * q^25 + 7 * q^29 + 11 * q^31 + q^35 + 3 * q^37 - 7 * q^41 + 4 * q^43 - 5 * q^47 + 2 * q^49 + 7 * q^53 - 12 * q^59 + 14 * q^61 + 18 * q^65 + 15 * q^67 - 9 * q^71 + 20 * q^73 - q^79 + 3 * q^83 + q^85 + 11 * q^89 + 3 * q^91 - 32 * q^95 - 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

−2.37228
3.37228

−2.37228

1.00000

1.2

3.37228

1.00000

Atkin-Lehner signs

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

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	1512.2.a.q	yes	2
3.b	odd	2	1		1512.2.a.n	✓	2
4.b	odd	2	1		3024.2.a.bl		2
12.b	even	2	1		3024.2.a.bf		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1512.2.a.n	✓	2	3.b	odd	2	1
1512.2.a.q	yes	2	1.a	even	1	1	trivial
3024.2.a.bf		2	12.b	even	2	1
3024.2.a.bl		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 $S_{2}^{\mathrm{new}}(\Gamma_0(1512))$ :

T_{5}^{2} - T_{5} - 8

T5^2 - T5 - 8

T_{11}

T11

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} - T - 8$ T^2 - T - 8
$7$	$(T - 1)^{2}$ (T - 1)^2
$11$	$T^{2}$ T^2
$13$	$T^{2} - 3T - 6$ T^2 - 3*T - 6
$17$	$(T - 1)^{2}$ (T - 1)^2
$19$	$T^{2} - 2T - 32$ T^2 - 2*T - 32
$23$	$T^{2} + 5T - 2$ T^2 + 5*T - 2
$29$	$T^{2} - 7T + 4$ T^2 - 7*T + 4
$31$	$T^{2} - 11T + 22$ T^2 - 11*T + 22
$37$	$T^{2} - 3T - 6$ T^2 - 3*T - 6
$41$	$T^{2} + 7T - 62$ T^2 + 7*T - 62
$43$	$T^{2} - 4T - 29$ T^2 - 4*T - 29
$47$	$T^{2} + 5T - 2$ T^2 + 5*T - 2
$53$	$T^{2} - 7T - 62$ T^2 - 7*T - 62
$59$	$T^{2} + 12T + 3$ T^2 + 12*T + 3
$61$	$T^{2} - 14T + 16$ T^2 - 14*T + 16
$67$	$T^{2} - 15T + 48$ T^2 - 15*T + 48
$71$	$T^{2} + 9T + 12$ T^2 + 9*T + 12
$73$	$(T - 10)^{2}$ (T - 10)^2
$79$	$T^{2} + T - 74$ T^2 + T - 74
$83$	$T^{2} - 3T - 204$ T^2 - 3*T - 204
$89$	$T^{2} - 11T + 22$ T^2 - 11*T + 22
$97$	$T^{2} + 4T - 128$ T^2 + 4*T - 128
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