# Properties

 Label 864.2.bf.a Level $864$ Weight $2$ Character orbit 864.bf Analytic conductor $6.899$ Analytic rank $0$ Dimension $204$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(49,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 14]))

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

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

chi := DirichletCharacter("864.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.bf (of order $$18$$, degree $$6$$, 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: $$6.89907473464$$ Analytic rank: $$0$$ Dimension: $$204$$ Relative dimension: $$34$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$204 q + 12 q^{7} - 12 q^{9}+O(q^{10})$$ 204 * q + 12 * q^7 - 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$204 q + 12 q^{7} - 12 q^{9} + 12 q^{15} - 6 q^{17} + 12 q^{23} - 12 q^{25} + 12 q^{31} + 12 q^{39} - 24 q^{41} + 12 q^{47} - 12 q^{49} + 24 q^{55} - 30 q^{57} + 72 q^{63} - 12 q^{65} + 90 q^{71} - 6 q^{73} + 12 q^{79} - 12 q^{81} + 48 q^{87} - 6 q^{89} + 42 q^{95} - 12 q^{97}+O(q^{100})$$ 204 * q + 12 * q^7 - 12 * q^9 + 12 * q^15 - 6 * q^17 + 12 * q^23 - 12 * q^25 + 12 * q^31 + 12 * q^39 - 24 * q^41 + 12 * q^47 - 12 * q^49 + 24 * q^55 - 30 * q^57 + 72 * q^63 - 12 * q^65 + 90 * q^71 - 6 * q^73 + 12 * q^79 - 12 * q^81 + 48 * q^87 - 6 * q^89 + 42 * q^95 - 12 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 0 −1.73005 0.0831268i 0 −0.689006 1.89303i 0 0.247544 + 1.40389i 0 2.98618 + 0.287628i 0
49.2 0 −1.69237 + 0.368629i 0 1.30726 + 3.59167i 0 0.406982 + 2.30811i 0 2.72823 1.24771i 0
49.3 0 −1.67311 0.447985i 0 0.141945 + 0.389992i 0 −0.275847 1.56441i 0 2.59862 + 1.49906i 0
49.4 0 −1.59613 + 0.672593i 0 −0.882298 2.42409i 0 −0.818935 4.64441i 0 2.09524 2.14709i 0
49.5 0 −1.59491 + 0.675483i 0 −0.545265 1.49810i 0 0.575033 + 3.26118i 0 2.08744 2.15466i 0
49.6 0 −1.50226 0.862098i 0 0.890128 + 2.44561i 0 0.478063 + 2.71123i 0 1.51358 + 2.59019i 0
49.7 0 −1.40645 1.01088i 0 −1.34230 3.68793i 0 0.499959 + 2.83541i 0 0.956226 + 2.84352i 0
49.8 0 −1.34903 + 1.08633i 0 1.01754 + 2.79566i 0 −0.349397 1.98153i 0 0.639780 2.93099i 0
49.9 0 −1.21028 + 1.23904i 0 −0.405536 1.11420i 0 −0.0413952 0.234764i 0 −0.0704471 2.99917i 0
49.10 0 −1.09709 1.34029i 0 0.0373388 + 0.102588i 0 −0.639213 3.62515i 0 −0.592777 + 2.94085i 0
49.11 0 −1.07385 1.35898i 0 −0.0465753 0.127965i 0 0.252128 + 1.42989i 0 −0.693675 + 2.91870i 0
49.12 0 −0.759610 + 1.55660i 0 0.857221 + 2.35519i 0 −0.442985 2.51229i 0 −1.84598 2.36481i 0
49.13 0 −0.684682 1.59098i 0 −1.15428 3.17134i 0 −0.593321 3.36489i 0 −2.06242 + 2.17863i 0
49.14 0 −0.375176 + 1.69093i 0 0.429535 + 1.18014i 0 0.899466 + 5.10113i 0 −2.71849 1.26879i 0
49.15 0 −0.368292 + 1.69244i 0 −0.355965 0.978005i 0 0.272085 + 1.54307i 0 −2.72872 1.24663i 0
49.16 0 −0.195652 1.72096i 0 1.34802 + 3.70366i 0 −0.00177522 0.0100678i 0 −2.92344 + 0.673422i 0
49.17 0 −0.123826 1.72762i 0 0.608623 + 1.67218i 0 −0.408085 2.31436i 0 −2.96933 + 0.427848i 0
49.18 0 0.123826 + 1.72762i 0 −0.608623 1.67218i 0 −0.408085 2.31436i 0 −2.96933 + 0.427848i 0
49.19 0 0.195652 + 1.72096i 0 −1.34802 3.70366i 0 −0.00177522 0.0100678i 0 −2.92344 + 0.673422i 0
49.20 0 0.368292 1.69244i 0 0.355965 + 0.978005i 0 0.272085 + 1.54307i 0 −2.72872 1.24663i 0
See next 80 embeddings (of 204 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 817.34 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.b even 2 1 inner
27.e even 9 1 inner
216.t even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.bf.a 204
4.b odd 2 1 216.2.t.a 204
8.b even 2 1 inner 864.2.bf.a 204
8.d odd 2 1 216.2.t.a 204
12.b even 2 1 648.2.t.a 204
24.f even 2 1 648.2.t.a 204
27.e even 9 1 inner 864.2.bf.a 204
108.j odd 18 1 216.2.t.a 204
108.l even 18 1 648.2.t.a 204
216.r odd 18 1 216.2.t.a 204
216.t even 18 1 inner 864.2.bf.a 204
216.v even 18 1 648.2.t.a 204

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.t.a 204 4.b odd 2 1
216.2.t.a 204 8.d odd 2 1
216.2.t.a 204 108.j odd 18 1
216.2.t.a 204 216.r odd 18 1
648.2.t.a 204 12.b even 2 1
648.2.t.a 204 24.f even 2 1
648.2.t.a 204 108.l even 18 1
648.2.t.a 204 216.v even 18 1
864.2.bf.a 204 1.a even 1 1 trivial
864.2.bf.a 204 8.b even 2 1 inner
864.2.bf.a 204 27.e even 9 1 inner
864.2.bf.a 204 216.t even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(864, [\chi])$$.