# Properties

 Label 864.2.w.a Level $864$ Weight $2$ Character orbit 864.w Analytic conductor $6.899$ Analytic rank $0$ Dimension $128$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(107,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([4, 5, 4]))

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

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

chi := DirichletCharacter("864.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 864.w (of order $$8$$, degree $$4$$, 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: $$128$$ Relative dimension: $$32$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$128 q+O(q^{10})$$ 128 * q $$\operatorname{Tr}(f)(q) =$$ $$128 q - 16 q^{10} + 32 q^{16} + 16 q^{22} - 32 q^{40} - 32 q^{46} + 16 q^{52} - 32 q^{55} - 32 q^{58} - 64 q^{61} - 48 q^{64} - 64 q^{67} + 96 q^{70} - 32 q^{76} + 64 q^{79} - 80 q^{82} - 80 q^{88} + 96 q^{91} - 144 q^{94}+O(q^{100})$$ 128 * q - 16 * q^10 + 32 * q^16 + 16 * q^22 - 32 * q^40 - 32 * q^46 + 16 * q^52 - 32 * q^55 - 32 * q^58 - 64 * q^61 - 48 * q^64 - 64 * q^67 + 96 * q^70 - 32 * q^76 + 64 * q^79 - 80 * q^82 - 80 * q^88 + 96 * q^91 - 144 * q^94

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
107.1 −1.41174 0.0836695i 0 1.98600 + 0.236238i −0.833551 0.345268i 0 −3.45115 + 3.45115i −2.78394 0.499674i 0 1.14787 + 0.557170i
107.2 −1.39320 0.242879i 0 1.88202 + 0.676758i 3.50739 + 1.45281i 0 −2.11049 + 2.11049i −2.45766 1.39996i 0 −4.53364 2.87593i
107.3 −1.37973 0.310377i 0 1.80733 + 0.856477i −1.77643 0.735821i 0 2.53483 2.53483i −2.22781 1.74266i 0 2.22262 + 1.56660i
107.4 −1.36834 + 0.357292i 0 1.74468 0.977791i 0.202568 + 0.0839066i 0 −0.0119413 + 0.0119413i −2.03796 + 1.96131i 0 −0.307161 0.0424363i
107.5 −1.25271 + 0.656291i 0 1.13856 1.64428i 1.92929 + 0.799136i 0 1.25938 1.25938i −0.347162 + 2.80704i 0 −2.94130 + 0.265087i
107.6 −1.19791 0.751667i 0 0.869994 + 1.80086i −3.00885 1.24631i 0 −0.981137 + 0.981137i 0.311473 2.81122i 0 2.66753 + 3.75462i
107.7 −1.07850 0.914792i 0 0.326309 + 1.97320i 1.10881 + 0.459283i 0 2.29035 2.29035i 1.45315 2.42660i 0 −0.775695 1.50966i
107.8 −1.01437 + 0.985425i 0 0.0578733 1.99916i −2.90620 1.20379i 0 −0.0493336 + 0.0493336i 1.91132 + 2.08491i 0 4.13418 1.64276i
107.9 −0.948429 1.04904i 0 −0.200965 + 1.98988i 1.78263 + 0.738390i 0 −0.622002 + 0.622002i 2.27806 1.67644i 0 −0.916100 2.57036i
107.10 −0.787598 + 1.17460i 0 −0.759380 1.85023i 1.54565 + 0.640231i 0 2.40953 2.40953i 2.77137 + 0.565266i 0 −1.96937 + 1.31128i
107.11 −0.649337 + 1.25633i 0 −1.15672 1.63156i 3.53205 + 1.46302i 0 −2.51075 + 2.51075i 2.80088 0.393792i 0 −4.13152 + 3.48742i
107.12 −0.501484 1.32231i 0 −1.49703 + 1.32624i 0.112585 + 0.0466342i 0 −1.45843 + 1.45843i 2.50444 + 1.31445i 0 0.00520548 0.172259i
107.13 −0.394008 1.35822i 0 −1.68952 + 1.07030i −2.44724 1.01368i 0 3.38875 3.38875i 2.11938 + 1.87302i 0 −0.412566 + 3.72328i
107.14 −0.167507 + 1.40426i 0 −1.94388 0.470445i −3.93413 1.62957i 0 1.80111 1.80111i 0.986240 2.65091i 0 2.94733 5.25157i
107.15 −0.0811033 1.41189i 0 −1.98684 + 0.229017i 1.02313 + 0.423795i 0 −0.549738 + 0.549738i 0.484486 + 2.78662i 0 0.515371 1.47892i
107.16 −0.0606141 + 1.41291i 0 −1.99265 0.171285i 1.65199 + 0.684276i 0 −1.93899 + 1.93899i 0.362794 2.80506i 0 −1.06696 + 2.29264i
107.17 0.0606141 1.41291i 0 −1.99265 0.171285i −1.65199 0.684276i 0 −1.93899 + 1.93899i −0.362794 + 2.80506i 0 −1.06696 + 2.29264i
107.18 0.0811033 + 1.41189i 0 −1.98684 + 0.229017i −1.02313 0.423795i 0 −0.549738 + 0.549738i −0.484486 2.78662i 0 0.515371 1.47892i
107.19 0.167507 1.40426i 0 −1.94388 0.470445i 3.93413 + 1.62957i 0 1.80111 1.80111i −0.986240 + 2.65091i 0 2.94733 5.25157i
107.20 0.394008 + 1.35822i 0 −1.68952 + 1.07030i 2.44724 + 1.01368i 0 3.38875 3.38875i −2.11938 1.87302i 0 −0.412566 + 3.72328i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 755.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.h odd 8 1 inner
96.o even 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.w.a 128
3.b odd 2 1 inner 864.2.w.a 128
32.h odd 8 1 inner 864.2.w.a 128
96.o even 8 1 inner 864.2.w.a 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.w.a 128 1.a even 1 1 trivial
864.2.w.a 128 3.b odd 2 1 inner
864.2.w.a 128 32.h odd 8 1 inner
864.2.w.a 128 96.o even 8 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{128} + 768 T_{5}^{122} + 201472 T_{5}^{120} + 269056 T_{5}^{118} + 294912 T_{5}^{116} + 72082176 T_{5}^{114} + 14577833472 T_{5}^{112} + 29671856640 T_{5}^{110} + 32138166272 T_{5}^{108} + \cdots + 22\!\cdots\!76$$ acting on $$S_{2}^{\mathrm{new}}(864, [\chi])$$.