Properties

Label 512.2.o.a
Level $512$
Weight $2$
Character orbit 512.o
Analytic conductor $4.088$
Analytic rank $0$
Dimension $4032$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [512,2,Mod(5,512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(128))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("512.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.o (of order \(128\), degree \(64\), 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: \(4.08834058349\)
Analytic rank: \(0\)
Dimension: \(4032\)
Relative dimension: \(63\) over \(\Q(\zeta_{128})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{128}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 4032 q - 64 q^{2} - 64 q^{3} - 64 q^{4} - 64 q^{5} - 64 q^{6} - 64 q^{7} - 64 q^{8} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 4032 q - 64 q^{2} - 64 q^{3} - 64 q^{4} - 64 q^{5} - 64 q^{6} - 64 q^{7} - 64 q^{8} - 64 q^{9} - 64 q^{10} - 64 q^{11} - 64 q^{12} - 64 q^{13} - 64 q^{14} - 64 q^{15} - 64 q^{16} - 64 q^{17} - 64 q^{18} - 64 q^{19} - 64 q^{20} - 64 q^{21} - 64 q^{22} - 64 q^{23} - 64 q^{24} - 64 q^{25} - 64 q^{26} - 64 q^{27} - 64 q^{28} - 64 q^{29} - 64 q^{30} - 64 q^{31} - 64 q^{32} - 64 q^{33} - 64 q^{34} - 64 q^{35} - 64 q^{36} - 64 q^{37} - 64 q^{38} - 64 q^{39} - 64 q^{40} - 64 q^{41} - 64 q^{42} - 64 q^{43} - 64 q^{44} - 64 q^{45} - 64 q^{46} - 64 q^{47} - 64 q^{48} - 64 q^{49} - 64 q^{50} - 64 q^{51} - 64 q^{52} - 64 q^{53} - 64 q^{54} - 64 q^{55} - 64 q^{56} - 64 q^{57} - 64 q^{58} - 64 q^{59} - 64 q^{60} - 64 q^{61} - 64 q^{62} - 64 q^{63} - 64 q^{64} - 64 q^{65} - 64 q^{66} - 64 q^{67} - 64 q^{68} - 64 q^{69} - 64 q^{70} - 64 q^{71} - 64 q^{72} - 64 q^{73} - 64 q^{74} - 64 q^{75} - 64 q^{76} - 64 q^{77} - 64 q^{78} - 64 q^{79} - 64 q^{80} - 64 q^{81} - 64 q^{82} - 64 q^{83} - 64 q^{84} - 64 q^{85} - 64 q^{86} - 64 q^{87} - 64 q^{88} - 64 q^{89} - 64 q^{90} - 64 q^{91} - 64 q^{92} - 64 q^{93} - 64 q^{94} - 64 q^{95} - 64 q^{96} - 64 q^{97} - 64 q^{98} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41230 0.0736323i 0.668640 + 0.775139i 1.98916 + 0.207981i −0.455024 0.0111702i −0.887242 1.14396i −0.579250 + 2.31250i −2.79396 0.440197i 0.286430 1.93095i 0.641806 + 0.0492801i
5.2 −1.40353 + 0.173508i 1.19537 + 1.38577i 1.93979 0.487048i 2.41010 + 0.0591645i −1.91818 1.73756i −0.464653 + 1.85500i −2.63804 + 1.02016i −0.0512479 + 0.345485i −3.39291 + 0.335133i
5.3 −1.38795 0.271261i −1.88159 2.18129i 1.85283 + 0.752997i −2.67923 0.0657714i 2.01986 + 3.53793i −0.581717 + 2.32235i −2.36739 1.54773i −0.777434 + 5.24104i 3.70081 + 0.818059i
5.4 −1.37702 + 0.322217i −0.556414 0.645038i 1.79235 0.887397i −2.40397 0.0590141i 0.974033 + 0.708942i 1.06676 4.25872i −2.18216 + 1.79949i 0.333714 2.24971i 3.32932 0.693336i
5.5 −1.37343 0.337181i −0.833389 0.966130i 1.77262 + 0.926190i 3.44138 + 0.0844812i 0.818841 + 1.60791i 0.359378 1.43472i −2.12227 1.86975i 0.201322 1.35721i −4.69801 1.27640i
5.6 −1.37064 0.348339i 0.943297 + 1.09354i 1.75732 + 0.954897i −1.67661 0.0411585i −0.911998 1.82744i 0.698599 2.78896i −2.07603 1.92096i 0.134164 0.904459i 2.28370 + 0.640444i
5.7 −1.34552 + 0.435416i −0.955526 1.10772i 1.62083 1.17172i −0.885780 0.0217447i 1.76799 + 1.07440i −0.000328152 0.00131006i −1.67066 + 2.28230i 0.126178 0.850621i 1.20130 0.356425i
5.8 −1.34363 + 0.441201i 1.62403 + 1.88270i 1.61068 1.18562i −3.80814 0.0934845i −3.01274 1.81313i −0.545363 + 2.17721i −1.64106 + 2.30367i −0.466898 + 3.14757i 5.15797 1.55455i
5.9 −1.21995 0.715356i −0.595169 0.689967i 0.976532 + 1.74539i 0.263295 + 0.00646353i 0.232502 + 1.26748i −0.849867 + 3.39286i 0.0572585 2.82785i 0.318364 2.14624i −0.316582 0.196235i
5.10 −1.19094 + 0.762675i 0.173232 + 0.200824i 0.836653 1.81659i 2.81168 + 0.0690230i −0.359472 0.107049i 0.330864 1.32088i 0.389072 + 2.80154i 0.429870 2.89795i −3.40118 + 2.06220i
5.11 −1.19074 0.762973i 2.25131 + 2.60990i 0.835745 + 1.81701i −1.49149 0.0366139i −0.689458 4.82541i 0.0527871 0.210738i 0.391171 2.80125i −1.30296 + 8.78387i 1.74804 + 1.18156i
5.12 −1.11786 + 0.866245i −1.12670 1.30616i 0.499239 1.93669i −2.53567 0.0622471i 2.39095 + 0.484108i −0.796848 + 3.18120i 1.11957 + 2.59742i 0.00359615 0.0242433i 2.88845 2.12692i
5.13 −1.09603 + 0.893716i −1.81468 2.10372i 0.402544 1.95907i 2.77109 + 0.0680264i 3.86907 + 0.683922i −0.160347 + 0.640139i 1.30965 + 2.50695i −0.692379 + 4.66764i −3.09798 + 2.40201i
5.14 −1.09005 0.900993i 1.45075 + 1.68182i 0.376424 + 1.96426i 2.75573 + 0.0676494i −0.0660818 3.14038i −0.901721 + 3.59987i 1.35946 2.48030i −0.283655 + 1.91225i −2.94294 2.55664i
5.15 −1.06151 0.934455i −1.75460 2.03407i 0.253587 + 1.98386i −0.223365 0.00548331i −0.0382295 + 3.79878i 0.548061 2.18798i 1.58464 2.34284i −0.618627 + 4.17044i 0.231980 + 0.214546i
5.16 −1.05109 + 0.946158i 1.45593 + 1.68782i 0.209569 1.98899i −0.0634026 0.00155645i −3.12725 0.396512i 1.07059 4.27405i 1.66162 + 2.28889i −0.288833 + 1.94716i 0.0681143 0.0583530i
5.17 −0.962532 1.03611i 0.360620 + 0.418058i −0.147066 + 1.99459i −3.88959 0.0954842i 0.0860484 0.776038i −0.0593750 + 0.237038i 2.20817 1.76748i 0.395465 2.66601i 3.64492 + 4.12197i
5.18 −0.901401 + 1.08971i 1.88802 + 2.18874i −0.374952 1.96454i 1.75891 + 0.0431789i −4.08696 + 0.0844684i −0.903089 + 3.60533i 2.47877 + 1.36225i −0.785769 + 5.29723i −1.63254 + 1.87779i
5.19 −0.890574 1.09858i 1.01161 + 1.17273i −0.413756 + 1.95673i 2.43657 + 0.0598145i 0.387431 2.15574i 0.840927 3.35717i 2.51811 1.28807i 0.0882358 0.594837i −2.10424 2.73004i
5.20 −0.744441 1.20242i −2.03236 2.35607i −0.891615 + 1.79026i 2.68864 + 0.0660025i −1.32001 + 4.19771i −0.738845 + 2.94963i 2.81639 0.260649i −0.980393 + 6.60927i −1.92217 3.28200i
See next 80 embeddings (of 4032 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.63
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
512.o even 128 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 512.2.o.a 4032
512.o even 128 1 inner 512.2.o.a 4032
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.o.a 4032 1.a even 1 1 trivial
512.2.o.a 4032 512.o even 128 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(512, [\chi])\).