# Properties

 Label 216.2.i.a Level $216$ Weight $2$ Character orbit 216.i Analytic conductor $1.725$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(73,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("216.73");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.i (of order $$3$$, degree $$2$$, not 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: $$1.72476868366$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} +O(q^{10})$$ q - z * q^5 + (-3*z + 3) * q^7 $$q - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + ( - 5 \zeta_{6} + 5) q^{11} + 5 \zeta_{6} q^{13} + 2 q^{17} - 4 q^{19} - \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + (9 \zeta_{6} - 9) q^{29} + \zeta_{6} q^{31} - 3 q^{35} - 6 q^{37} + 3 \zeta_{6} q^{41} + (\zeta_{6} - 1) q^{43} + (3 \zeta_{6} - 3) q^{47} - 2 \zeta_{6} q^{49} - 2 q^{53} - 5 q^{55} + 11 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} + ( - 5 \zeta_{6} + 5) q^{65} + \zeta_{6} q^{67} - 4 q^{71} - 2 q^{73} - 15 \zeta_{6} q^{77} + (\zeta_{6} - 1) q^{79} + ( - \zeta_{6} + 1) q^{83} - 2 \zeta_{6} q^{85} + 18 q^{89} + 15 q^{91} + 4 \zeta_{6} q^{95} + ( - 13 \zeta_{6} + 13) q^{97} +O(q^{100})$$ q - z * q^5 + (-3*z + 3) * q^7 + (-5*z + 5) * q^11 + 5*z * q^13 + 2 * q^17 - 4 * q^19 - z * q^23 + (-4*z + 4) * q^25 + (9*z - 9) * q^29 + z * q^31 - 3 * q^35 - 6 * q^37 + 3*z * q^41 + (z - 1) * q^43 + (3*z - 3) * q^47 - 2*z * q^49 - 2 * q^53 - 5 * q^55 + 11*z * q^59 + (7*z - 7) * q^61 + (-5*z + 5) * q^65 + z * q^67 - 4 * q^71 - 2 * q^73 - 15*z * q^77 + (z - 1) * q^79 + (-z + 1) * q^83 - 2*z * q^85 + 18 * q^89 + 15 * q^91 + 4*z * q^95 + (-13*z + 13) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{5} + 3 q^{7}+O(q^{10})$$ 2 * q - q^5 + 3 * q^7 $$2 q - q^{5} + 3 q^{7} + 5 q^{11} + 5 q^{13} + 4 q^{17} - 8 q^{19} - q^{23} + 4 q^{25} - 9 q^{29} + q^{31} - 6 q^{35} - 12 q^{37} + 3 q^{41} - q^{43} - 3 q^{47} - 2 q^{49} - 4 q^{53} - 10 q^{55} + 11 q^{59} - 7 q^{61} + 5 q^{65} + q^{67} - 8 q^{71} - 4 q^{73} - 15 q^{77} - q^{79} + q^{83} - 2 q^{85} + 36 q^{89} + 30 q^{91} + 4 q^{95} + 13 q^{97}+O(q^{100})$$ 2 * q - q^5 + 3 * q^7 + 5 * q^11 + 5 * q^13 + 4 * q^17 - 8 * q^19 - q^23 + 4 * q^25 - 9 * q^29 + q^31 - 6 * q^35 - 12 * q^37 + 3 * q^41 - q^43 - 3 * q^47 - 2 * q^49 - 4 * q^53 - 10 * q^55 + 11 * q^59 - 7 * q^61 + 5 * q^65 + q^67 - 8 * q^71 - 4 * q^73 - 15 * q^77 - q^79 + q^83 - 2 * q^85 + 36 * q^89 + 30 * q^91 + 4 * q^95 + 13 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$$.

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −0.500000 0.866025i 0 1.50000 2.59808i 0 0 0
145.1 0 0 0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.2.i.a 2
3.b odd 2 1 72.2.i.a 2
4.b odd 2 1 432.2.i.a 2
8.b even 2 1 1728.2.i.h 2
8.d odd 2 1 1728.2.i.g 2
9.c even 3 1 inner 216.2.i.a 2
9.c even 3 1 648.2.a.c 1
9.d odd 6 1 72.2.i.a 2
9.d odd 6 1 648.2.a.a 1
12.b even 2 1 144.2.i.b 2
24.f even 2 1 576.2.i.c 2
24.h odd 2 1 576.2.i.d 2
36.f odd 6 1 432.2.i.a 2
36.f odd 6 1 1296.2.a.i 1
36.h even 6 1 144.2.i.b 2
36.h even 6 1 1296.2.a.e 1
72.j odd 6 1 576.2.i.d 2
72.j odd 6 1 5184.2.a.s 1
72.l even 6 1 576.2.i.c 2
72.l even 6 1 5184.2.a.x 1
72.n even 6 1 1728.2.i.h 2
72.n even 6 1 5184.2.a.i 1
72.p odd 6 1 1728.2.i.g 2
72.p odd 6 1 5184.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 3.b odd 2 1
72.2.i.a 2 9.d odd 6 1
144.2.i.b 2 12.b even 2 1
144.2.i.b 2 36.h even 6 1
216.2.i.a 2 1.a even 1 1 trivial
216.2.i.a 2 9.c even 3 1 inner
432.2.i.a 2 4.b odd 2 1
432.2.i.a 2 36.f odd 6 1
576.2.i.c 2 24.f even 2 1
576.2.i.c 2 72.l even 6 1
576.2.i.d 2 24.h odd 2 1
576.2.i.d 2 72.j odd 6 1
648.2.a.a 1 9.d odd 6 1
648.2.a.c 1 9.c even 3 1
1296.2.a.e 1 36.h even 6 1
1296.2.a.i 1 36.f odd 6 1
1728.2.i.g 2 8.d odd 2 1
1728.2.i.g 2 72.p odd 6 1
1728.2.i.h 2 8.b even 2 1
1728.2.i.h 2 72.n even 6 1
5184.2.a.i 1 72.n even 6 1
5184.2.a.n 1 72.p odd 6 1
5184.2.a.s 1 72.j odd 6 1
5184.2.a.x 1 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2} - 3T + 9$$
$11$ $$T^{2} - 5T + 25$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + T + 1$$
$29$ $$T^{2} + 9T + 81$$
$31$ $$T^{2} - T + 1$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} + 3T + 9$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 11T + 121$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - T + 1$$
$71$ $$(T + 4)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + T + 1$$
$83$ $$T^{2} - T + 1$$
$89$ $$(T - 18)^{2}$$
$97$ $$T^{2} - 13T + 169$$