LMFDB - Newform orbit 9360.2.a.ci

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1560)
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.

$f(q)$	$=$	$q - q^{5} + \beta q^{7} + (\beta - 4) q^{11} + q^{13} + ( - 3 \beta + 2) q^{17} + 2 \beta q^{19} + (\beta - 4) q^{23} + q^{25} + 2 q^{29} + ( - 2 \beta + 4) q^{31} - \beta q^{35} + ( - 5 \beta + 2) q^{37} + \cdots + (5 \beta - 10) q^{97} +O(q^{100})$ q - q^5 + b * q^7 + (b - 4) * q^11 + q^13 + (-3b + 2) q^17 + 2b q^19 + (b - 4) * q^23 + q^25 + 2 * q^29 + (-2b + 4) q^31 - b * q^35 + (-5b + 2) q^37 + (3b + 2) q^41 + (-4b + 4) q^43 + (-2b + 4) q^47 + (b - 3) * q^49 + (b - 2) * q^53 + (-b + 4) * q^55 - 8 * q^59 + (3b - 6) q^61 - q^65 + (-4b + 8) q^67 - 3b q^71 + (-4b + 10) q^73 + (-3b + 4) q^77 + (5b - 4) q^79 - 8 * q^83 + (3b - 2) q^85 + (-3b + 10) q^89 + b * q^91 - 2b q^95 + (5b - 10) q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{5} + q^{7} - 7 q^{11} + 2 q^{13} + q^{17} + 2 q^{19} - 7 q^{23} + 2 q^{25} + 4 q^{29} + 6 q^{31} - q^{35} - q^{37} + 7 q^{41} + 4 q^{43} + 6 q^{47} - 5 q^{49} - 3 q^{53} + 7 q^{55} - 16 q^{59}+ \cdots - 15 q^{97}+O(q^{100})$ 2 * q - 2 * q^5 + q^7 - 7 * q^11 + 2 * q^13 + q^17 + 2 * q^19 - 7 * q^23 + 2 * q^25 + 4 * q^29 + 6 * q^31 - q^35 - q^37 + 7 * q^41 + 4 * q^43 + 6 * q^47 - 5 * q^49 - 3 * q^53 + 7 * q^55 - 16 * q^59 - 9 * q^61 - 2 * q^65 + 12 * q^67 - 3 * q^71 + 16 * q^73 + 5 * q^77 - 3 * q^79 - 16 * q^83 - q^85 + 17 * q^89 + q^91 - 2 * q^95 - 15 * 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

−1.56155
2.56155

−1.00000

−1.56155

1.2

−1.00000

2.56155

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$3$	$-1$
$5$	$+1$
$13$	$-1$

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	9360.2.a.ci		2
3.b	odd	2	1		3120.2.a.bg		2
4.b	odd	2	1		4680.2.a.y		2
12.b	even	2	1		1560.2.a.o	✓	2
60.h	even	2	1		7800.2.a.bd		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1560.2.a.o	✓	2	12.b	even	2	1
3120.2.a.bg		2	3.b	odd	2	1
4680.2.a.y		2	4.b	odd	2	1
7800.2.a.bd		2	60.h	even	2	1
9360.2.a.ci		2	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(9360))$ :

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

T_{11}^{2} + 7T_{11} + 8

T_{17}^{2} - T_{17} - 38

T_{19}^{2} - 2T_{19} - 16

T_{31}^{2} - 6T_{31} - 8

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T + 1)^{2}$ (T + 1)^2
$7$	$T^{2} - T - 4$ T^2 - T - 4
$11$	$T^{2} + 7T + 8$ T^2 + 7*T + 8
$13$	$(T - 1)^{2}$ (T - 1)^2
$17$	$T^{2} - T - 38$ T^2 - T - 38
$19$	$T^{2} - 2T - 16$ T^2 - 2*T - 16
$23$	$T^{2} + 7T + 8$ T^2 + 7*T + 8
$29$	$(T - 2)^{2}$ (T - 2)^2
$31$	$T^{2} - 6T - 8$ T^2 - 6*T - 8
$37$	$T^{2} + T - 106$ T^2 + T - 106
$41$	$T^{2} - 7T - 26$ T^2 - 7*T - 26
$43$	$T^{2} - 4T - 64$ T^2 - 4*T - 64
$47$	$T^{2} - 6T - 8$ T^2 - 6*T - 8
$53$	$T^{2} + 3T - 2$ T^2 + 3*T - 2
$59$	$(T + 8)^{2}$ (T + 8)^2
$61$	$T^{2} + 9T - 18$ T^2 + 9*T - 18
$67$	$T^{2} - 12T - 32$ T^2 - 12*T - 32
$71$	$T^{2} + 3T - 36$ T^2 + 3*T - 36
$73$	$T^{2} - 16T - 4$ T^2 - 16*T - 4
$79$	$T^{2} + 3T - 104$ T^2 + 3*T - 104
$83$	$(T + 8)^{2}$ (T + 8)^2
$89$	$T^{2} - 17T + 34$ T^2 - 17*T + 34
$97$	$T^{2} + 15T - 50$ T^2 + 15*T - 50
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