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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 $4$

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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.

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

Display 100 coefficients 10 coefficients

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:

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 \(S_{2}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display