Properties

Label 819.2.k.a
Level $819$
Weight $2$
Character orbit 819.k
Analytic conductor $6.540$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(529,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.k (of order \(3\), degree \(2\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(108\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - q^{2} + q^{3} - 105 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{7} - 12 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - q^{2} + q^{3} - 105 q^{4} + 2 q^{5} - 6 q^{6} - 6 q^{7} - 12 q^{8} - q^{9} + 6 q^{10} + 10 q^{11} + 12 q^{12} - 3 q^{13} + 6 q^{14} + 13 q^{15} - 99 q^{16} - 36 q^{17} - 6 q^{18} - 28 q^{20} + q^{21} - 6 q^{22} + 3 q^{23} + 16 q^{24} - 90 q^{25} - 8 q^{26} - 14 q^{27} + 24 q^{28} - 2 q^{29} + 28 q^{30} - 12 q^{31} + 11 q^{32} - 29 q^{33} - 6 q^{34} - 13 q^{35} - 8 q^{36} - 6 q^{37} + 40 q^{38} - q^{39} + 24 q^{40} - 4 q^{41} - 24 q^{42} - 12 q^{43} - 14 q^{44} - 22 q^{45} + 12 q^{46} + 8 q^{47} - 72 q^{48} - 6 q^{49} - 10 q^{50} + 24 q^{51} + 9 q^{52} + 2 q^{53} - 12 q^{54} - 9 q^{55} - 8 q^{56} - 26 q^{57} + 18 q^{58} - q^{59} - 26 q^{60} + 6 q^{61} + 32 q^{62} - 31 q^{63} + 156 q^{64} + 11 q^{65} - 13 q^{66} + 6 q^{67} + 35 q^{68} + 29 q^{69} + 6 q^{70} - 4 q^{71} + 37 q^{72} - 18 q^{73} - q^{74} - 29 q^{75} - 12 q^{76} - 12 q^{77} + 2 q^{78} - 18 q^{79} + 21 q^{80} + 23 q^{81} + 6 q^{82} + 10 q^{83} - 60 q^{84} + 3 q^{85} - 4 q^{86} - 44 q^{87} + 42 q^{88} + 6 q^{89} - 60 q^{90} - 12 q^{91} - 4 q^{92} - 18 q^{93} - 9 q^{94} - 15 q^{95} + 90 q^{96} - 6 q^{97} + 106 q^{98} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 −1.37149 + 2.37548i 1.32649 1.11374i −2.76195 4.78383i 0.561780 0.973031i 0.826421 + 4.67854i 0.457055 + 2.60597i 9.66594 0.519145 2.95474i 1.54095 + 2.66900i
529.2 −1.36329 + 2.36128i −1.63790 0.563285i −2.71710 4.70615i 0.883550 1.53035i 3.56300 3.09962i 0.667122 2.56026i 9.36359 2.36542 + 1.84521i 2.40906 + 4.17262i
529.3 −1.36199 + 2.35903i −0.101015 + 1.72910i −2.71002 4.69389i −1.44364 + 2.50046i −3.94142 2.59331i −1.95628 1.78129i 9.31608 −2.97959 0.349332i −3.93244 6.81119i
529.4 −1.32017 + 2.28661i −1.68127 + 0.416334i −2.48572 4.30540i −0.208519 + 0.361166i 1.26757 4.39404i −0.741997 + 2.53957i 7.84566 2.65333 1.39994i −0.550563 0.953603i
529.5 −1.31849 + 2.28369i 1.51170 + 0.845443i −2.47684 4.29001i −0.480093 + 0.831546i −3.92389 + 2.33754i 2.60436 + 0.466161i 7.78878 1.57045 + 2.55611i −1.26600 2.19277i
529.6 −1.30037 + 2.25231i 0.362376 + 1.69372i −2.38192 4.12560i 1.31473 2.27718i −4.28599 1.38628i 1.55247 2.14239i 7.18802 −2.73737 + 1.22753i 3.41927 + 5.92236i
529.7 −1.22812 + 2.12717i 1.69033 0.377876i −2.01658 3.49282i −0.826620 + 1.43175i −1.27213 + 4.05970i −2.00295 1.72864i 4.99394 2.71442 1.27747i −2.03038 3.51673i
529.8 −1.22804 + 2.12703i −1.22282 + 1.22666i −2.01616 3.49209i 1.11119 1.92464i −1.10747 4.10737i −2.33013 + 1.25320i 4.99154 −0.00941039 2.99999i 2.72917 + 4.72707i
529.9 −1.22619 + 2.12382i 1.66268 + 0.485288i −2.00707 3.47635i 1.85349 3.21033i −3.06942 + 2.93617i −2.64307 0.118995i 4.93944 2.52899 + 1.61376i 4.54545 + 7.87294i
529.10 −1.21225 + 2.09968i −0.832368 1.51893i −1.93909 3.35861i 0.0986359 0.170842i 4.19831 + 0.0936229i −2.20595 + 1.46074i 4.55367 −1.61433 + 2.52863i 0.239142 + 0.414207i
529.11 −1.20426 + 2.08585i 0.373598 1.69128i −1.90051 3.29177i 1.63962 2.83991i 3.07784 + 2.81602i 1.94181 1.79704i 4.33779 −2.72085 1.26372i 3.94908 + 6.84001i
529.12 −1.13226 + 1.96114i −1.42832 + 0.979753i −1.56404 2.70900i −0.598097 + 1.03594i −0.304201 3.91046i 2.56564 0.646143i 2.55458 1.08017 2.79879i −1.35441 2.34590i
529.13 −1.10549 + 1.91477i −0.461989 1.66930i −1.44423 2.50148i 0.515659 0.893148i 3.70705 + 0.960797i 2.26346 + 1.36994i 1.96438 −2.57313 + 1.54240i 1.14012 + 1.97474i
529.14 −1.09960 + 1.90456i 1.55431 + 0.764270i −1.41824 2.45646i −1.70739 + 2.95729i −3.16472 + 2.11990i −1.55946 + 2.13731i 1.83958 1.83178 + 2.37583i −3.75490 6.50367i
529.15 −1.09934 + 1.90410i 1.36680 1.06388i −1.41708 2.45445i −1.08428 + 1.87802i 0.523170 + 3.77210i 1.78621 1.95178i 1.83403 0.736304 2.90824i −2.38397 4.12915i
529.16 −1.09502 + 1.89664i 0.394274 1.68658i −1.39816 2.42168i −1.84829 + 3.20134i 2.76709 + 2.59464i −1.50421 + 2.17654i 1.74397 −2.68910 1.32995i −4.04786 7.01109i
529.17 −1.08646 + 1.88181i 0.418879 + 1.68064i −1.36081 2.35699i 1.60386 2.77798i −3.61774 1.03770i 0.267891 + 2.63215i 1.56801 −2.64908 + 1.40797i 3.48508 + 6.03634i
529.18 −1.08286 + 1.87557i −1.11275 1.32732i −1.34516 2.32989i −0.323241 + 0.559870i 3.69443 0.649729i −1.75235 1.98224i 1.49505 −0.523581 + 2.95396i −0.700049 1.21252i
529.19 −1.07426 + 1.86067i −1.71786 + 0.221230i −1.30806 2.26563i −2.10940 + 3.65359i 1.43380 3.43404i −2.00111 1.73077i 1.32376 2.90211 0.760085i −4.53208 7.84980i
529.20 −0.975545 + 1.68969i 0.232549 + 1.71637i −0.903376 1.56469i −0.367459 + 0.636458i −3.12700 1.28146i 0.390144 + 2.61683i −0.377046 −2.89184 + 0.798280i −0.716946 1.24179i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
819.k even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.k.a 216
7.c even 3 1 819.2.u.a yes 216
9.c even 3 1 819.2.p.a yes 216
13.c even 3 1 819.2.l.a yes 216
63.h even 3 1 819.2.l.a yes 216
91.h even 3 1 819.2.p.a yes 216
117.f even 3 1 819.2.u.a yes 216
819.k even 3 1 inner 819.2.k.a 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.k.a 216 1.a even 1 1 trivial
819.2.k.a 216 819.k even 3 1 inner
819.2.l.a yes 216 13.c even 3 1
819.2.l.a yes 216 63.h even 3 1
819.2.p.a yes 216 9.c even 3 1
819.2.p.a yes 216 91.h even 3 1
819.2.u.a yes 216 7.c even 3 1
819.2.u.a yes 216 117.f even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).