# Properties

 Label 216.3.m.a Level $216$ Weight $3$ Character orbit 216.m Analytic conductor $5.886$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,3,Mod(17,216)]

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, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("216.17");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 216.m (of order $$6$$, 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: $$5.88557371018$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 72) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{7}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^5 + (-b3 - 3*b2 - b1) * q^7 $$q + ( - \beta_{2} + \beta_1 - 1) q^{5} + ( - \beta_{3} - 3 \beta_{2} - \beta_1) q^{7} + ( - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 6) q^{11} + ( - 2 \beta_{3} + 7 \beta_{2} + \beta_1 - 7) q^{13} + ( - 2 \beta_{3} + 16 \beta_{2} - 8) q^{17} + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{19} + ( - 5 \beta_{2} + \beta_1 - 5) q^{23} + ( - 2 \beta_{3} + 10 \beta_{2} - 2 \beta_1) q^{25} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{29} + (2 \beta_{3} - 37 \beta_{2} - \beta_1 + 37) q^{31} + ( - 58 \beta_{2} + 29) q^{35} + ( - 2 \beta_{3} + 4 \beta_1 - 30) q^{37} + (23 \beta_{2} + 23) q^{41} + ( - 4 \beta_{3} + 5 \beta_{2} - 4 \beta_1) q^{43} + (3 \beta_{3} + 29 \beta_{2} - 3 \beta_1 - 58) q^{47} + (12 \beta_{3} + 56 \beta_{2} - 6 \beta_1 - 56) q^{49} + (8 \beta_{3} + 64 \beta_{2} - 32) q^{53} + ( - \beta_{3} + 2 \beta_1 + 55) q^{55} + (3 \beta_{2} - 2 \beta_1 + 3) q^{59} + ( - \beta_{3} - 31 \beta_{2} - \beta_1) q^{61} + (10 \beta_{3} - 39 \beta_{2} - 10 \beta_1 + 78) q^{65} + (20 \beta_{3} + 11 \beta_{2} - 10 \beta_1 - 11) q^{67} + ( - 2 \beta_{3} - 48 \beta_{2} + 24) q^{71} + ( - 6 \beta_{3} + 12 \beta_1 + 10) q^{73} + ( - 73 \beta_{2} - 15 \beta_1 - 73) q^{77} + (7 \beta_{3} - 43 \beta_{2} + 7 \beta_1) q^{79} + (14 \beta_{3} - 11 \beta_{2} - 14 \beta_1 + 22) q^{83} + (20 \beta_{3} - 88 \beta_{2} - 10 \beta_1 + 88) q^{85} + ( - 6 \beta_{3} + 48 \beta_{2} - 24) q^{89} + ( - 4 \beta_{3} + 8 \beta_1 - 75) q^{91} + (62 \beta_{2} - 4 \beta_1 + 62) q^{95} + ( - 2 \beta_{3} + 121 \beta_{2} - 2 \beta_1) q^{97}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^5 + (-b3 - 3*b2 - b1) * q^7 + (-2*b3 - 3*b2 + 2*b1 + 6) * q^11 + (-2*b3 + 7*b2 + b1 - 7) * q^13 + (-2*b3 + 16*b2 - 8) * q^17 + (-2*b3 + 4*b1 + 2) * q^19 + (-5*b2 + b1 - 5) * q^23 + (-2*b3 + 10*b2 - 2*b1) * q^25 + (-3*b3 + b2 + 3*b1 - 2) * q^29 + (2*b3 - 37*b2 - b1 + 37) * q^31 + (-58*b2 + 29) * q^35 + (-2*b3 + 4*b1 - 30) * q^37 + (23*b2 + 23) * q^41 + (-4*b3 + 5*b2 - 4*b1) * q^43 + (3*b3 + 29*b2 - 3*b1 - 58) * q^47 + (12*b3 + 56*b2 - 6*b1 - 56) * q^49 + (8*b3 + 64*b2 - 32) * q^53 + (-b3 + 2*b1 + 55) * q^55 + (3*b2 - 2*b1 + 3) * q^59 + (-b3 - 31*b2 - b1) * q^61 + (10*b3 - 39*b2 - 10*b1 + 78) * q^65 + (20*b3 + 11*b2 - 10*b1 - 11) * q^67 + (-2*b3 - 48*b2 + 24) * q^71 + (-6*b3 + 12*b1 + 10) * q^73 + (-73*b2 - 15*b1 - 73) * q^77 + (7*b3 - 43*b2 + 7*b1) * q^79 + (14*b3 - 11*b2 - 14*b1 + 22) * q^83 + (20*b3 - 88*b2 - 10*b1 + 88) * q^85 + (-6*b3 + 48*b2 - 24) * q^89 + (-4*b3 + 8*b1 - 75) * q^91 + (62*b2 - 4*b1 + 62) * q^95 + (-2*b3 + 121*b2 - 2*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{5} - 6 q^{7}+O(q^{10})$$ 4 * q - 6 * q^5 - 6 * q^7 $$4 q - 6 q^{5} - 6 q^{7} + 18 q^{11} - 14 q^{13} + 8 q^{19} - 30 q^{23} + 20 q^{25} - 6 q^{29} + 74 q^{31} - 120 q^{37} + 138 q^{41} + 10 q^{43} - 174 q^{47} - 112 q^{49} + 220 q^{55} + 18 q^{59} - 62 q^{61} + 234 q^{65} - 22 q^{67} + 40 q^{73} - 438 q^{77} - 86 q^{79} + 66 q^{83} + 176 q^{85} - 300 q^{91} + 372 q^{95} + 242 q^{97}+O(q^{100})$$ 4 * q - 6 * q^5 - 6 * q^7 + 18 * q^11 - 14 * q^13 + 8 * q^19 - 30 * q^23 + 20 * q^25 - 6 * q^29 + 74 * q^31 - 120 * q^37 + 138 * q^41 + 10 * q^43 - 174 * q^47 - 112 * q^49 + 220 * q^55 + 18 * q^59 - 62 * q^61 + 234 * q^65 - 22 * q^67 + 40 * q^73 - 438 * q^77 - 86 * q^79 + 66 * q^83 + 176 * q^85 - 300 * q^91 + 372 * q^95 + 242 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$4\nu$$ 4*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$2\nu^{3}$$ 2*v^3
 $$\nu$$ $$=$$ $$( \beta_1 ) / 4$$ (b1) / 4 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$55$$ $$109$$ $$137$$ $$\chi(n)$$ $$1$$ $$1$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −1.22474 − 0.707107i 1.22474 + 0.707107i −1.22474 + 0.707107i 1.22474 − 0.707107i
0 0 0 −6.39898 3.69445i 0 3.39898 + 5.88721i 0 0 0
17.2 0 0 0 3.39898 + 1.96240i 0 −6.39898 11.0834i 0 0 0
89.1 0 0 0 −6.39898 + 3.69445i 0 3.39898 5.88721i 0 0 0
89.2 0 0 0 3.39898 1.96240i 0 −6.39898 + 11.0834i 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.m.a 4
3.b odd 2 1 72.3.m.a 4
4.b odd 2 1 432.3.q.c 4
8.b even 2 1 1728.3.q.e 4
8.d odd 2 1 1728.3.q.f 4
9.c even 3 1 72.3.m.a 4
9.c even 3 1 648.3.e.b 4
9.d odd 6 1 inner 216.3.m.a 4
9.d odd 6 1 648.3.e.b 4
12.b even 2 1 144.3.q.d 4
24.f even 2 1 576.3.q.c 4
24.h odd 2 1 576.3.q.h 4
36.f odd 6 1 144.3.q.d 4
36.f odd 6 1 1296.3.e.c 4
36.h even 6 1 432.3.q.c 4
36.h even 6 1 1296.3.e.c 4
72.j odd 6 1 1728.3.q.e 4
72.l even 6 1 1728.3.q.f 4
72.n even 6 1 576.3.q.h 4
72.p odd 6 1 576.3.q.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.a 4 3.b odd 2 1
72.3.m.a 4 9.c even 3 1
144.3.q.d 4 12.b even 2 1
144.3.q.d 4 36.f odd 6 1
216.3.m.a 4 1.a even 1 1 trivial
216.3.m.a 4 9.d odd 6 1 inner
432.3.q.c 4 4.b odd 2 1
432.3.q.c 4 36.h even 6 1
576.3.q.c 4 24.f even 2 1
576.3.q.c 4 72.p odd 6 1
576.3.q.h 4 24.h odd 2 1
576.3.q.h 4 72.n even 6 1
648.3.e.b 4 9.c even 3 1
648.3.e.b 4 9.d odd 6 1
1296.3.e.c 4 36.f odd 6 1
1296.3.e.c 4 36.h even 6 1
1728.3.q.e 4 8.b even 2 1
1728.3.q.e 4 72.j odd 6 1
1728.3.q.f 4 8.d odd 2 1
1728.3.q.f 4 72.l even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 6T_{5}^{3} - 17T_{5}^{2} - 174T_{5} + 841$$ acting on $$S_{3}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} - 17 T^{2} - 174 T + 841$$
$7$ $$T^{4} + 6 T^{3} + 123 T^{2} + \cdots + 7569$$
$11$ $$T^{4} - 18 T^{3} + 7 T^{2} + \cdots + 10201$$
$13$ $$T^{4} + 14 T^{3} + 243 T^{2} + \cdots + 2209$$
$17$ $$T^{4} + 640T^{2} + 4096$$
$19$ $$(T^{2} - 4 T - 380)^{2}$$
$23$ $$T^{4} + 30 T^{3} + 343 T^{2} + \cdots + 1849$$
$29$ $$T^{4} + 6 T^{3} - 273 T^{2} + \cdots + 81225$$
$31$ $$T^{4} - 74 T^{3} + 4203 T^{2} + \cdots + 1620529$$
$37$ $$(T^{2} + 60 T + 516)^{2}$$
$41$ $$(T^{2} - 69 T + 1587)^{2}$$
$43$ $$T^{4} - 10 T^{3} + 1611 T^{2} + \cdots + 2283121$$
$47$ $$T^{4} + 174 T^{3} + 12327 T^{2} + \cdots + 4995225$$
$53$ $$T^{4} + 10240 T^{2} + \cdots + 1048576$$
$59$ $$T^{4} - 18 T^{3} + 7 T^{2} + \cdots + 10201$$
$61$ $$T^{4} + 62 T^{3} + 2979 T^{2} + \cdots + 748225$$
$67$ $$T^{4} + 22 T^{3} + 9963 T^{2} + \cdots + 89851441$$
$71$ $$T^{4} + 3712 T^{2} + \cdots + 2560000$$
$73$ $$(T^{2} - 20 T - 3356)^{2}$$
$79$ $$T^{4} + 86 T^{3} + 10251 T^{2} + \cdots + 8151025$$
$83$ $$T^{4} - 66 T^{3} - 4457 T^{2} + \cdots + 34916281$$
$89$ $$T^{4} + 5760 T^{2} + 331776$$
$97$ $$T^{4} - 242 T^{3} + \cdots + 203262049$$