# Properties

 Label 216.2.a.a Level $216$ Weight $2$ Character orbit 216.a Self dual yes Analytic conductor $1.725$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,2,Mod(1,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("216.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 216.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: $$1.72476868366$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - 4 q^{5} - 3 q^{7}+O(q^{10})$$ q - 4 * q^5 - 3 * q^7 $$q - 4 q^{5} - 3 q^{7} - 4 q^{11} + q^{13} + 4 q^{17} - q^{19} - 4 q^{23} + 11 q^{25} - 4 q^{31} + 12 q^{35} - 9 q^{37} - 8 q^{43} + 12 q^{47} + 2 q^{49} + 8 q^{53} + 16 q^{55} - 4 q^{59} - 5 q^{61} - 4 q^{65} + 11 q^{67} - 8 q^{71} + q^{73} + 12 q^{77} - 5 q^{79} - 8 q^{83} - 16 q^{85} - 12 q^{89} - 3 q^{91} + 4 q^{95} + 5 q^{97}+O(q^{100})$$ q - 4 * q^5 - 3 * q^7 - 4 * q^11 + q^13 + 4 * q^17 - q^19 - 4 * q^23 + 11 * q^25 - 4 * q^31 + 12 * q^35 - 9 * q^37 - 8 * q^43 + 12 * q^47 + 2 * q^49 + 8 * q^53 + 16 * q^55 - 4 * q^59 - 5 * q^61 - 4 * q^65 + 11 * q^67 - 8 * q^71 + q^73 + 12 * q^77 - 5 * q^79 - 8 * q^83 - 16 * q^85 - 12 * q^89 - 3 * q^91 + 4 * q^95 + 5 * 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
 0
0 0 0 −4.00000 0 −3.00000 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
$$2$$ $$1$$
$$3$$ $$1$$

## 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 216.2.a.a 1
3.b odd 2 1 216.2.a.d yes 1
4.b odd 2 1 432.2.a.a 1
5.b even 2 1 5400.2.a.bn 1
5.c odd 4 2 5400.2.f.e 2
8.b even 2 1 1728.2.a.ba 1
8.d odd 2 1 1728.2.a.bb 1
9.c even 3 2 648.2.i.h 2
9.d odd 6 2 648.2.i.a 2
12.b even 2 1 432.2.a.h 1
15.d odd 2 1 5400.2.a.bp 1
15.e even 4 2 5400.2.f.v 2
24.f even 2 1 1728.2.a.b 1
24.h odd 2 1 1728.2.a.a 1
36.f odd 6 2 1296.2.i.q 2
36.h even 6 2 1296.2.i.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.a.a 1 1.a even 1 1 trivial
216.2.a.d yes 1 3.b odd 2 1
432.2.a.a 1 4.b odd 2 1
432.2.a.h 1 12.b even 2 1
648.2.i.a 2 9.d odd 6 2
648.2.i.h 2 9.c even 3 2
1296.2.i.a 2 36.h even 6 2
1296.2.i.q 2 36.f odd 6 2
1728.2.a.a 1 24.h odd 2 1
1728.2.a.b 1 24.f even 2 1
1728.2.a.ba 1 8.b even 2 1
1728.2.a.bb 1 8.d odd 2 1
5400.2.a.bn 1 5.b even 2 1
5400.2.a.bp 1 15.d odd 2 1
5400.2.f.e 2 5.c odd 4 2
5400.2.f.v 2 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(216))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 4$$
$7$ $$T + 3$$
$11$ $$T + 4$$
$13$ $$T - 1$$
$17$ $$T - 4$$
$19$ $$T + 1$$
$23$ $$T + 4$$
$29$ $$T$$
$31$ $$T + 4$$
$37$ $$T + 9$$
$41$ $$T$$
$43$ $$T + 8$$
$47$ $$T - 12$$
$53$ $$T - 8$$
$59$ $$T + 4$$
$61$ $$T + 5$$
$67$ $$T - 11$$
$71$ $$T + 8$$
$73$ $$T - 1$$
$79$ $$T + 5$$
$83$ $$T + 8$$
$89$ $$T + 12$$
$97$ $$T - 5$$