# Properties

 Label 216.1.p.a Level $216$ Weight $1$ Character orbit 216.p Analytic conductor $0.108$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -8 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,1,Mod(19,216)]

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([3, 3, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("216.19");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 216.p (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: $$0.107798042729$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 72) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of 12.0.139314069504.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{8} +O(q^{10})$$ q - z^2 * q^2 - z * q^4 - q^8 $$q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - q^{8} + \zeta_{6}^{2} q^{11} + \zeta_{6}^{2} q^{16} + q^{17} - q^{19} + \zeta_{6} q^{22} + \zeta_{6}^{2} q^{25} + \zeta_{6} q^{32} - \zeta_{6}^{2} q^{34} + \zeta_{6}^{2} q^{38} - \zeta_{6} q^{41} - \zeta_{6}^{2} q^{43} + q^{44} - \zeta_{6} q^{49} + \zeta_{6} q^{50} - \zeta_{6} q^{59} + q^{64} + \zeta_{6} q^{67} - \zeta_{6} q^{68} - q^{73} + \zeta_{6} q^{76} - q^{82} - \zeta_{6}^{2} q^{83} - \zeta_{6} q^{86} - \zeta_{6}^{2} q^{88} - q^{89} - \zeta_{6}^{2} q^{97} - q^{98} +O(q^{100})$$ q - z^2 * q^2 - z * q^4 - q^8 + z^2 * q^11 + z^2 * q^16 + q^17 - q^19 + z * q^22 + z^2 * q^25 + z * q^32 - z^2 * q^34 + z^2 * q^38 - z * q^41 - z^2 * q^43 + q^44 - z * q^49 + z * q^50 - z * q^59 + q^64 + z * q^67 - z * q^68 - q^73 + z * q^76 - q^82 - z^2 * q^83 - z * q^86 - z^2 * q^88 - q^89 - z^2 * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - 2 q^{8}+O(q^{10})$$ 2 * q + q^2 - q^4 - 2 * q^8 $$2 q + q^{2} - q^{4} - 2 q^{8} - q^{11} - q^{16} + 2 q^{17} - 2 q^{19} + q^{22} - q^{25} + q^{32} + q^{34} - q^{38} - q^{41} + q^{43} + 2 q^{44} - q^{49} + q^{50} - q^{59} + 2 q^{64} + q^{67} - q^{68} - 2 q^{73} + q^{76} - 2 q^{82} + 2 q^{83} - q^{86} + q^{88} - 4 q^{89} + q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 - q^4 - 2 * q^8 - q^11 - q^16 + 2 * q^17 - 2 * q^19 + q^22 - q^25 + q^32 + q^34 - q^38 - q^41 + q^43 + 2 * q^44 - q^49 + q^50 - q^59 + 2 * q^64 + q^67 - q^68 - 2 * q^73 + q^76 - 2 * q^82 + 2 * q^83 - q^86 + q^88 - 4 * q^89 + q^97 - 2 * q^98

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 −1.00000 0 0
91.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 −1.00000 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.c even 3 1 inner
72.p odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.1.p.a 2
3.b odd 2 1 72.1.p.a 2
4.b odd 2 1 864.1.t.a 2
8.b even 2 1 864.1.t.a 2
8.d odd 2 1 CM 216.1.p.a 2
9.c even 3 1 inner 216.1.p.a 2
9.c even 3 1 648.1.b.a 1
9.d odd 6 1 72.1.p.a 2
9.d odd 6 1 648.1.b.b 1
12.b even 2 1 288.1.t.a 2
15.d odd 2 1 1800.1.bk.d 2
15.e even 4 2 1800.1.ba.b 4
21.c even 2 1 3528.1.cg.a 2
21.g even 6 1 3528.1.ba.a 2
21.g even 6 1 3528.1.ce.b 2
21.h odd 6 1 3528.1.ba.b 2
21.h odd 6 1 3528.1.ce.a 2
24.f even 2 1 72.1.p.a 2
24.h odd 2 1 288.1.t.a 2
36.f odd 6 1 864.1.t.a 2
36.f odd 6 1 2592.1.b.a 1
36.h even 6 1 288.1.t.a 2
36.h even 6 1 2592.1.b.b 1
45.h odd 6 1 1800.1.bk.d 2
45.l even 12 2 1800.1.ba.b 4
48.i odd 4 2 2304.1.o.c 4
48.k even 4 2 2304.1.o.c 4
63.i even 6 1 3528.1.ba.a 2
63.j odd 6 1 3528.1.ba.b 2
63.n odd 6 1 3528.1.ce.a 2
63.o even 6 1 3528.1.cg.a 2
63.s even 6 1 3528.1.ce.b 2
72.j odd 6 1 288.1.t.a 2
72.j odd 6 1 2592.1.b.b 1
72.l even 6 1 72.1.p.a 2
72.l even 6 1 648.1.b.b 1
72.n even 6 1 864.1.t.a 2
72.n even 6 1 2592.1.b.a 1
72.p odd 6 1 inner 216.1.p.a 2
72.p odd 6 1 648.1.b.a 1
120.m even 2 1 1800.1.bk.d 2
120.q odd 4 2 1800.1.ba.b 4
144.u even 12 2 2304.1.o.c 4
144.w odd 12 2 2304.1.o.c 4
168.e odd 2 1 3528.1.cg.a 2
168.v even 6 1 3528.1.ba.b 2
168.v even 6 1 3528.1.ce.a 2
168.be odd 6 1 3528.1.ba.a 2
168.be odd 6 1 3528.1.ce.b 2
360.bd even 6 1 1800.1.bk.d 2
360.bt odd 12 2 1800.1.ba.b 4
504.u odd 6 1 3528.1.ce.b 2
504.bt even 6 1 3528.1.ba.b 2
504.cm odd 6 1 3528.1.ba.a 2
504.co odd 6 1 3528.1.cg.a 2
504.cy even 6 1 3528.1.ce.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 3.b odd 2 1
72.1.p.a 2 9.d odd 6 1
72.1.p.a 2 24.f even 2 1
72.1.p.a 2 72.l even 6 1
216.1.p.a 2 1.a even 1 1 trivial
216.1.p.a 2 8.d odd 2 1 CM
216.1.p.a 2 9.c even 3 1 inner
216.1.p.a 2 72.p odd 6 1 inner
288.1.t.a 2 12.b even 2 1
288.1.t.a 2 24.h odd 2 1
288.1.t.a 2 36.h even 6 1
288.1.t.a 2 72.j odd 6 1
648.1.b.a 1 9.c even 3 1
648.1.b.a 1 72.p odd 6 1
648.1.b.b 1 9.d odd 6 1
648.1.b.b 1 72.l even 6 1
864.1.t.a 2 4.b odd 2 1
864.1.t.a 2 8.b even 2 1
864.1.t.a 2 36.f odd 6 1
864.1.t.a 2 72.n even 6 1
1800.1.ba.b 4 15.e even 4 2
1800.1.ba.b 4 45.l even 12 2
1800.1.ba.b 4 120.q odd 4 2
1800.1.ba.b 4 360.bt odd 12 2
1800.1.bk.d 2 15.d odd 2 1
1800.1.bk.d 2 45.h odd 6 1
1800.1.bk.d 2 120.m even 2 1
1800.1.bk.d 2 360.bd even 6 1
2304.1.o.c 4 48.i odd 4 2
2304.1.o.c 4 48.k even 4 2
2304.1.o.c 4 144.u even 12 2
2304.1.o.c 4 144.w odd 12 2
2592.1.b.a 1 36.f odd 6 1
2592.1.b.a 1 72.n even 6 1
2592.1.b.b 1 36.h even 6 1
2592.1.b.b 1 72.j odd 6 1
3528.1.ba.a 2 21.g even 6 1
3528.1.ba.a 2 63.i even 6 1
3528.1.ba.a 2 168.be odd 6 1
3528.1.ba.a 2 504.cm odd 6 1
3528.1.ba.b 2 21.h odd 6 1
3528.1.ba.b 2 63.j odd 6 1
3528.1.ba.b 2 168.v even 6 1
3528.1.ba.b 2 504.bt even 6 1
3528.1.ce.a 2 21.h odd 6 1
3528.1.ce.a 2 63.n odd 6 1
3528.1.ce.a 2 168.v even 6 1
3528.1.ce.a 2 504.cy even 6 1
3528.1.ce.b 2 21.g even 6 1
3528.1.ce.b 2 63.s even 6 1
3528.1.ce.b 2 168.be odd 6 1
3528.1.ce.b 2 504.u odd 6 1
3528.1.cg.a 2 21.c even 2 1
3528.1.cg.a 2 63.o even 6 1
3528.1.cg.a 2 168.e odd 2 1
3528.1.cg.a 2 504.co odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(216, [\chi])$$.

## Hecke characteristic polynomials

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