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.
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) |
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 \(S_{2}^{\mathrm{new}}(819, [\chi])\).