LMFDB - Newform orbit 936.2.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(1,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("936.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$936 = 2^{3} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	936.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:	$7.47399762919$
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 104)
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.

$f(q)$	$=$	$q + ( - \beta - 1) q^{5} + (\beta - 1) q^{7} + ( - 2 \beta + 2) q^{11} + q^{13} + (3 \beta - 1) q^{17} + ( - 2 \beta + 2) q^{19} + 8 q^{23} + 3 \beta q^{25} + 2 q^{29} + 4 q^{31} + ( - \beta - 3) q^{35}+ \cdots + (4 \beta - 2) q^{97}+O(q^{100})$ q + (-b - 1) * q^5 + (b - 1) * q^7 + (-2b + 2) q^11 + q^13 + (3b - 1) q^17 + (-2b + 2) q^19 + 8 * q^23 + 3b q^25 + 2 * q^29 + 4 * q^31 + (-b - 3) * q^35 + (-3b + 5) q^37 - 2b q^41 + (b + 7) * q^43 + (3b + 5) q^47 + (-b - 2) * q^49 + 2b q^53 + (2b + 6) q^55 + (2b - 2) q^59 + (-2b + 8) q^61 + (-b - 1) * q^65 + (2b - 2) q^67 + (-3b + 3) q^71 - 6 * q^73 + (2b - 10) q^77 + 8 * q^79 + (4b + 4) q^83 + (-5b - 11) q^85 - 10 * q^89 + (b - 1) * q^91 + (2b + 6) q^95 + (4b - 2) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{5} - q^{7} + 2 q^{11} + 2 q^{13} + q^{17} + 2 q^{19} + 16 q^{23} + 3 q^{25} + 4 q^{29} + 8 q^{31} - 7 q^{35} + 7 q^{37} - 2 q^{41} + 15 q^{43} + 13 q^{47} - 5 q^{49} + 2 q^{53} + 14 q^{55}+ \cdots + 14 q^{95}+O(q^{100})$ 2 * q - 3 * q^5 - q^7 + 2 * q^11 + 2 * q^13 + q^17 + 2 * q^19 + 16 * q^23 + 3 * q^25 + 4 * q^29 + 8 * q^31 - 7 * q^35 + 7 * q^37 - 2 * q^41 + 15 * q^43 + 13 * q^47 - 5 * q^49 + 2 * q^53 + 14 * q^55 - 2 * q^59 + 14 * q^61 - 3 * q^65 - 2 * q^67 + 3 * q^71 - 12 * q^73 - 18 * q^77 + 16 * q^79 + 12 * q^83 - 27 * q^85 - 20 * q^89 - q^91 + 14 * q^95

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

−3.56155

1.56155

1.2

0.561553

−2.56155

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$-1$
$13$	$-1$

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	936.2.a.j		2
3.b	odd	2	1		104.2.a.b	✓	2
4.b	odd	2	1		1872.2.a.u		2
8.b	even	2	1		7488.2.a.cu		2
8.d	odd	2	1		7488.2.a.cv		2
12.b	even	2	1		208.2.a.e		2
15.d	odd	2	1		2600.2.a.p		2
15.e	even	4	2		2600.2.d.k		4
21.c	even	2	1		5096.2.a.m		2
24.f	even	2	1		832.2.a.n		2
24.h	odd	2	1		832.2.a.k		2
39.d	odd	2	1		1352.2.a.g		2
39.f	even	4	2		1352.2.f.c		4
39.h	odd	6	2		1352.2.i.d		4
39.i	odd	6	2		1352.2.i.f		4
39.k	even	12	4		1352.2.o.d		8
48.i	odd	4	2		3328.2.b.y		4
48.k	even	4	2		3328.2.b.w		4
60.h	even	2	1		5200.2.a.bw		2
156.h	even	2	1		2704.2.a.p		2
156.l	odd	4	2		2704.2.f.k		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.a.b	✓	2	3.b	odd	2	1
208.2.a.e		2	12.b	even	2	1
832.2.a.k		2	24.h	odd	2	1
832.2.a.n		2	24.f	even	2	1
936.2.a.j		2	1.a	even	1	1	trivial
1352.2.a.g		2	39.d	odd	2	1
1352.2.f.c		4	39.f	even	4	2
1352.2.i.d		4	39.h	odd	6	2
1352.2.i.f		4	39.i	odd	6	2
1352.2.o.d		8	39.k	even	12	4
1872.2.a.u		2	4.b	odd	2	1
2600.2.a.p		2	15.d	odd	2	1
2600.2.d.k		4	15.e	even	4	2
2704.2.a.p		2	156.h	even	2	1
2704.2.f.k		4	156.l	odd	4	2
3328.2.b.w		4	48.k	even	4	2
3328.2.b.y		4	48.i	odd	4	2
5096.2.a.m		2	21.c	even	2	1
5200.2.a.bw		2	60.h	even	2	1
7488.2.a.cu		2	8.b	even	2	1
7488.2.a.cv		2	8.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 $S_{2}^{\mathrm{new}}(\Gamma_0(936))$ :

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

T_{7}^{2} + T_{7} - 4

T_{11}^{2} - 2T_{11} - 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$T^{2} + 3T - 2$ T^2 + 3*T - 2
$7$	$T^{2} + T - 4$ T^2 + T - 4
$11$	$T^{2} - 2T - 16$ T^2 - 2*T - 16
$13$	$(T - 1)^{2}$ (T - 1)^2
$17$	$T^{2} - T - 38$ T^2 - T - 38
$19$	$T^{2} - 2T - 16$ T^2 - 2*T - 16
$23$	$(T - 8)^{2}$ (T - 8)^2
$29$	$(T - 2)^{2}$ (T - 2)^2
$31$	$(T - 4)^{2}$ (T - 4)^2
$37$	$T^{2} - 7T - 26$ T^2 - 7*T - 26
$41$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$43$	$T^{2} - 15T + 52$ T^2 - 15*T + 52
$47$	$T^{2} - 13T + 4$ T^2 - 13*T + 4
$53$	$T^{2} - 2T - 16$ T^2 - 2*T - 16
$59$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$61$	$T^{2} - 14T + 32$ T^2 - 14*T + 32
$67$	$T^{2} + 2T - 16$ T^2 + 2*T - 16
$71$	$T^{2} - 3T - 36$ T^2 - 3*T - 36
$73$	$(T + 6)^{2}$ (T + 6)^2
$79$	$(T - 8)^{2}$ (T - 8)^2
$83$	$T^{2} - 12T - 32$ T^2 - 12*T - 32
$89$	$(T + 10)^{2}$ (T + 10)^2
$97$	$T^{2} - 68$ T^2 - 68
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