Properties

 Label 2160.2.h.e Level $2160$ Weight $2$ Character orbit 2160.h Analytic conductor $17.248$ Analytic rank $0$ Dimension $4$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,2,Mod(431,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2160 = 2^{4} \cdot 3^{3} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2160.h (of order $$2$$, degree $$1$$, minimal)

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: no Analytic conductor: $$17.2476868366$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{5}+O(q^{10})$$ q + b1 * q^5 $$q + \beta_1 q^{5} + 2 \beta_{3} q^{11} + 4 q^{13} - 3 \beta_1 q^{17} + 3 \beta_{2} q^{19} - \beta_{3} q^{23} - q^{25} + \beta_{2} q^{31} + 2 q^{37} - 6 \beta_1 q^{41} - 2 \beta_{2} q^{43} - 2 \beta_{3} q^{47} + 7 q^{49} + 3 \beta_1 q^{53} + 2 \beta_{2} q^{55} - 4 \beta_{3} q^{59} + 11 q^{61} + 4 \beta_1 q^{65} + 2 \beta_{2} q^{67} + 2 \beta_{3} q^{71} + 10 q^{73} + 7 \beta_{2} q^{79} + 3 \beta_{3} q^{83} + 3 q^{85} + 18 \beta_1 q^{89} - 3 \beta_{3} q^{95} + 16 q^{97}+O(q^{100})$$ q + b1 * q^5 + 2*b3 * q^11 + 4 * q^13 - 3*b1 * q^17 + 3*b2 * q^19 - b3 * q^23 - q^25 + b2 * q^31 + 2 * q^37 - 6*b1 * q^41 - 2*b2 * q^43 - 2*b3 * q^47 + 7 * q^49 + 3*b1 * q^53 + 2*b2 * q^55 - 4*b3 * q^59 + 11 * q^61 + 4*b1 * q^65 + 2*b2 * q^67 + 2*b3 * q^71 + 10 * q^73 + 7*b2 * q^79 + 3*b3 * q^83 + 3 * q^85 + 18*b1 * q^89 - 3*b3 * q^95 + 16 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 16 q^{13} - 4 q^{25} + 8 q^{37} + 28 q^{49} + 44 q^{61} + 40 q^{73} + 12 q^{85} + 64 q^{97}+O(q^{100})$$ 4 * q + 16 * q^13 - 4 * q^25 + 8 * q^37 + 28 * q^49 + 44 * q^61 + 40 * q^73 + 12 * q^85 + 64 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$1297$$ $$1621$$ $$2081$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
431.1
 −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i 0.866025 + 0.500000i
0 0 0 1.00000i 0 0 0 0 0
431.2 0 0 0 1.00000i 0 0 0 0 0
431.3 0 0 0 1.00000i 0 0 0 0 0
431.4 0 0 0 1.00000i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.h.e 4
3.b odd 2 1 inner 2160.2.h.e 4
4.b odd 2 1 inner 2160.2.h.e 4
12.b even 2 1 inner 2160.2.h.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.2.h.e 4 1.a even 1 1 trivial
2160.2.h.e 4 3.b odd 2 1 inner
2160.2.h.e 4 4.b odd 2 1 inner
2160.2.h.e 4 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2160, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} - 12$$ T11^2 - 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 12)^{2}$$
$13$ $$(T - 4)^{4}$$
$17$ $$(T^{2} + 9)^{2}$$
$19$ $$(T^{2} + 27)^{2}$$
$23$ $$(T^{2} - 3)^{2}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$(T - 2)^{4}$$
$41$ $$(T^{2} + 36)^{2}$$
$43$ $$(T^{2} + 12)^{2}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} - 48)^{2}$$
$61$ $$(T - 11)^{4}$$
$67$ $$(T^{2} + 12)^{2}$$
$71$ $$(T^{2} - 12)^{2}$$
$73$ $$(T - 10)^{4}$$
$79$ $$(T^{2} + 147)^{2}$$
$83$ $$(T^{2} - 27)^{2}$$
$89$ $$(T^{2} + 324)^{2}$$
$97$ $$(T - 16)^{4}$$