# Properties

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

# 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5} + (\beta_{2} + 3 \beta_1) q^{7}+O(q^{10})$$ q - b1 * q^5 + (b2 + 3*b1) * q^7 $$q - \beta_1 q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + ( - \beta_{3} - 3) q^{11} + (3 \beta_{3} + 1) q^{13} - 3 \beta_{2} q^{17} + (2 \beta_{2} - 3 \beta_1) q^{19} + ( - \beta_{3} + 6) q^{23} - q^{25} + (3 \beta_{2} + 3 \beta_1) q^{29} + (2 \beta_{2} - 3 \beta_1) q^{31} + (\beta_{3} + 3) q^{35} + 2 q^{37} + (3 \beta_{2} - 3 \beta_1) q^{41} + (5 \beta_{2} - 3 \beta_1) q^{43} + ( - 2 \beta_{3} - 6) q^{47} + ( - 6 \beta_{3} - 5) q^{49} + (3 \beta_{2} + 6 \beta_1) q^{53} + (\beta_{2} + 3 \beta_1) q^{55} + (5 \beta_{3} - 3) q^{59} - q^{61} + ( - 3 \beta_{2} - \beta_1) q^{65} + 6 \beta_{2} q^{67} + ( - \beta_{3} - 3) q^{71} + ( - 3 \beta_{3} + 1) q^{73} + ( - 6 \beta_{2} - 12 \beta_1) q^{77} + (4 \beta_{2} + 3 \beta_1) q^{79} - 3 \beta_{3} q^{83} - 3 \beta_{3} q^{85} + (3 \beta_{2} + 9 \beta_1) q^{89} + (10 \beta_{2} + 12 \beta_1) q^{91} + (2 \beta_{3} - 3) q^{95} - 8 q^{97}+O(q^{100})$$ q - b1 * q^5 + (b2 + 3*b1) * q^7 + (-b3 - 3) * q^11 + (3*b3 + 1) * q^13 - 3*b2 * q^17 + (2*b2 - 3*b1) * q^19 + (-b3 + 6) * q^23 - q^25 + (3*b2 + 3*b1) * q^29 + (2*b2 - 3*b1) * q^31 + (b3 + 3) * q^35 + 2 * q^37 + (3*b2 - 3*b1) * q^41 + (5*b2 - 3*b1) * q^43 + (-2*b3 - 6) * q^47 + (-6*b3 - 5) * q^49 + (3*b2 + 6*b1) * q^53 + (b2 + 3*b1) * q^55 + (5*b3 - 3) * q^59 - q^61 + (-3*b2 - b1) * q^65 + 6*b2 * q^67 + (-b3 - 3) * q^71 + (-3*b3 + 1) * q^73 + (-6*b2 - 12*b1) * q^77 + (4*b2 + 3*b1) * q^79 - 3*b3 * q^83 - 3*b3 * q^85 + (3*b2 + 9*b1) * q^89 + (10*b2 + 12*b1) * q^91 + (2*b3 - 3) * q^95 - 8 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{11} + 4 q^{13} + 24 q^{23} - 4 q^{25} + 12 q^{35} + 8 q^{37} - 24 q^{47} - 20 q^{49} - 12 q^{59} - 4 q^{61} - 12 q^{71} + 4 q^{73} - 12 q^{95} - 32 q^{97}+O(q^{100})$$ 4 * q - 12 * q^11 + 4 * q^13 + 24 * q^23 - 4 * q^25 + 12 * q^35 + 8 * q^37 - 24 * q^47 - 20 * q^49 - 12 * q^59 - 4 * q^61 - 12 * q^71 + 4 * q^73 - 12 * q^95 - 32 * 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 1.26795i 0 0 0
431.2 0 0 0 1.00000i 0 4.73205i 0 0 0
431.3 0 0 0 1.00000i 0 4.73205i 0 0 0
431.4 0 0 0 1.00000i 0 1.26795i 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
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.a 4
3.b odd 2 1 2160.2.h.f yes 4
4.b odd 2 1 2160.2.h.f yes 4
12.b even 2 1 inner 2160.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.2.h.a 4 1.a even 1 1 trivial
2160.2.h.a 4 12.b even 2 1 inner
2160.2.h.f yes 4 3.b odd 2 1
2160.2.h.f yes 4 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 $$S_{2}^{\mathrm{new}}(2160, [\chi])$$:

 $$T_{7}^{4} + 24T_{7}^{2} + 36$$ T7^4 + 24*T7^2 + 36 $$T_{11}^{2} + 6T_{11} + 6$$ T11^2 + 6*T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} + 24T^{2} + 36$$
$11$ $$(T^{2} + 6 T + 6)^{2}$$
$13$ $$(T^{2} - 2 T - 26)^{2}$$
$17$ $$(T^{2} + 27)^{2}$$
$19$ $$T^{4} + 42T^{2} + 9$$
$23$ $$(T^{2} - 12 T + 33)^{2}$$
$29$ $$T^{4} + 72T^{2} + 324$$
$31$ $$T^{4} + 42T^{2} + 9$$
$37$ $$(T - 2)^{4}$$
$41$ $$T^{4} + 72T^{2} + 324$$
$43$ $$T^{4} + 168T^{2} + 4356$$
$47$ $$(T^{2} + 12 T + 24)^{2}$$
$53$ $$T^{4} + 126T^{2} + 81$$
$59$ $$(T^{2} + 6 T - 66)^{2}$$
$61$ $$(T + 1)^{4}$$
$67$ $$(T^{2} + 108)^{2}$$
$71$ $$(T^{2} + 6 T + 6)^{2}$$
$73$ $$(T^{2} - 2 T - 26)^{2}$$
$79$ $$T^{4} + 114T^{2} + 1521$$
$83$ $$(T^{2} - 27)^{2}$$
$89$ $$T^{4} + 216T^{2} + 2916$$
$97$ $$(T + 8)^{4}$$