Properties

 Label 1080.2.a.m Level $1080$ Weight $2$ Character orbit 1080.a Self dual yes Analytic conductor $8.624$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(1,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, 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("1080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1080 = 2^{3} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1080.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: $$8.62384341830$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 18$$ x^2 - x - 18 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{5} + \beta q^{7}+O(q^{10})$$ q - q^5 + b * q^7 $$q - q^{5} + \beta q^{7} + ( - \beta + 1) q^{11} + 3 q^{13} + (\beta - 3) q^{17} + ( - \beta + 2) q^{19} + (\beta + 3) q^{23} + q^{25} + (\beta - 1) q^{29} + ( - \beta + 7) q^{31} - \beta q^{35} + ( - \beta + 4) q^{37} - 2 \beta q^{41} + ( - \beta + 3) q^{43} + (\beta + 1) q^{47} + (\beta + 11) q^{49} - 2 \beta q^{53} + (\beta - 1) q^{55} + 12 q^{59} + (\beta - 4) q^{61} - 3 q^{65} + (\beta + 10) q^{67} + (2 \beta + 4) q^{71} + \beta q^{73} - 18 q^{77} - 5 q^{79} + 6 q^{83} + ( - \beta + 3) q^{85} + 8 q^{89} + 3 \beta q^{91} + (\beta - 2) q^{95} + ( - 3 \beta - 4) q^{97} +O(q^{100})$$ q - q^5 + b * q^7 + (-b + 1) * q^11 + 3 * q^13 + (b - 3) * q^17 + (-b + 2) * q^19 + (b + 3) * q^23 + q^25 + (b - 1) * q^29 + (-b + 7) * q^31 - b * q^35 + (-b + 4) * q^37 - 2*b * q^41 + (-b + 3) * q^43 + (b + 1) * q^47 + (b + 11) * q^49 - 2*b * q^53 + (b - 1) * q^55 + 12 * q^59 + (b - 4) * q^61 - 3 * q^65 + (b + 10) * q^67 + (2*b + 4) * q^71 + b * q^73 - 18 * q^77 - 5 * q^79 + 6 * q^83 + (-b + 3) * q^85 + 8 * q^89 + 3*b * q^91 + (b - 2) * q^95 + (-3*b - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + q^7 $$2 q - 2 q^{5} + q^{7} + q^{11} + 6 q^{13} - 5 q^{17} + 3 q^{19} + 7 q^{23} + 2 q^{25} - q^{29} + 13 q^{31} - q^{35} + 7 q^{37} - 2 q^{41} + 5 q^{43} + 3 q^{47} + 23 q^{49} - 2 q^{53} - q^{55} + 24 q^{59} - 7 q^{61} - 6 q^{65} + 21 q^{67} + 10 q^{71} + q^{73} - 36 q^{77} - 10 q^{79} + 12 q^{83} + 5 q^{85} + 16 q^{89} + 3 q^{91} - 3 q^{95} - 11 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + q^7 + q^11 + 6 * q^13 - 5 * q^17 + 3 * q^19 + 7 * q^23 + 2 * q^25 - q^29 + 13 * q^31 - q^35 + 7 * q^37 - 2 * q^41 + 5 * q^43 + 3 * q^47 + 23 * q^49 - 2 * q^53 - q^55 + 24 * q^59 - 7 * q^61 - 6 * q^65 + 21 * q^67 + 10 * q^71 + q^73 - 36 * q^77 - 10 * q^79 + 12 * q^83 + 5 * q^85 + 16 * q^89 + 3 * q^91 - 3 * q^95 - 11 * 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
 −3.77200 4.77200
0 0 0 −1.00000 0 −3.77200 0 0 0
1.2 0 0 0 −1.00000 0 4.77200 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

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 1080.2.a.m 2
3.b odd 2 1 1080.2.a.n yes 2
4.b odd 2 1 2160.2.a.z 2
5.b even 2 1 5400.2.a.cb 2
5.c odd 4 2 5400.2.f.be 4
8.b even 2 1 8640.2.a.de 2
8.d odd 2 1 8640.2.a.db 2
9.c even 3 2 3240.2.q.bc 4
9.d odd 6 2 3240.2.q.z 4
12.b even 2 1 2160.2.a.bb 2
15.d odd 2 1 5400.2.a.ca 2
15.e even 4 2 5400.2.f.bd 4
24.f even 2 1 8640.2.a.cn 2
24.h odd 2 1 8640.2.a.cq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.a.m 2 1.a even 1 1 trivial
1080.2.a.n yes 2 3.b odd 2 1
2160.2.a.z 2 4.b odd 2 1
2160.2.a.bb 2 12.b even 2 1
3240.2.q.z 4 9.d odd 6 2
3240.2.q.bc 4 9.c even 3 2
5400.2.a.ca 2 15.d odd 2 1
5400.2.a.cb 2 5.b even 2 1
5400.2.f.bd 4 15.e even 4 2
5400.2.f.be 4 5.c odd 4 2
8640.2.a.cn 2 24.f even 2 1
8640.2.a.cq 2 24.h odd 2 1
8640.2.a.db 2 8.d odd 2 1
8640.2.a.de 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1080))$$:

 $$T_{7}^{2} - T_{7} - 18$$ T7^2 - T7 - 18 $$T_{11}^{2} - T_{11} - 18$$ T11^2 - T11 - 18 $$T_{17}^{2} + 5T_{17} - 12$$ T17^2 + 5*T17 - 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 18$$
$11$ $$T^{2} - T - 18$$
$13$ $$(T - 3)^{2}$$
$17$ $$T^{2} + 5T - 12$$
$19$ $$T^{2} - 3T - 16$$
$23$ $$T^{2} - 7T - 6$$
$29$ $$T^{2} + T - 18$$
$31$ $$T^{2} - 13T + 24$$
$37$ $$T^{2} - 7T - 6$$
$41$ $$T^{2} + 2T - 72$$
$43$ $$T^{2} - 5T - 12$$
$47$ $$T^{2} - 3T - 16$$
$53$ $$T^{2} + 2T - 72$$
$59$ $$(T - 12)^{2}$$
$61$ $$T^{2} + 7T - 6$$
$67$ $$T^{2} - 21T + 92$$
$71$ $$T^{2} - 10T - 48$$
$73$ $$T^{2} - T - 18$$
$79$ $$(T + 5)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 11T - 134$$