Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [729,2,Mod(4,729)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(729, base_ring=CyclotomicField(486))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("729.4");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 729 = 3^{6} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 729.k (of order \(243\), degree \(162\), 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: | \(5.82109430735\) |
Analytic rank: | \(0\) |
Dimension: | \(12960\) |
Relative dimension: | \(80\) over \(\Q(\zeta_{243})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{243}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
4.1 | −2.72490 | − | 0.0352304i | −1.51002 | − | 0.848437i | 5.42450 | + | 0.140291i | 0.633611 | + | 0.194162i | 4.08476 | + | 2.36510i | 1.09035 | − | 2.71893i | −9.33012 | − | 0.362051i | 1.56031 | + | 2.56231i | −1.71968 | − | 0.551395i |
4.2 | −2.68619 | − | 0.0347299i | 0.113106 | − | 1.72835i | 5.21506 | + | 0.134874i | −1.58849 | − | 0.486773i | −0.363851 | + | 4.63875i | −0.858532 | + | 2.14086i | −8.63515 | − | 0.335083i | −2.97441 | − | 0.390976i | 4.25007 | + | 1.36273i |
4.3 | −2.65153 | − | 0.0342818i | 0.952364 | + | 1.44672i | 5.03008 | + | 0.130090i | −0.525411 | − | 0.161006i | −2.47562 | − | 3.86867i | −0.732244 | + | 1.82595i | −8.03343 | − | 0.311734i | −1.18601 | + | 2.75561i | 1.38762 | + | 0.444923i |
4.4 | −2.61310 | − | 0.0337849i | 1.70554 | − | 0.301886i | 4.82780 | + | 0.124859i | −2.59518 | − | 0.795260i | −4.46694 | + | 0.731236i | 0.853432 | − | 2.12814i | −7.38859 | − | 0.286711i | 2.81773 | − | 1.02976i | 6.75458 | + | 2.16577i |
4.5 | −2.51518 | − | 0.0325190i | 1.53758 | − | 0.797395i | 4.32575 | + | 0.111875i | 3.23632 | + | 0.991730i | −3.89323 | + | 1.95559i | −1.26529 | + | 3.15518i | −5.84940 | − | 0.226983i | 1.72832 | − | 2.45212i | −8.10768 | − | 2.59962i |
4.6 | −2.48775 | − | 0.0321643i | 0.111243 | + | 1.72847i | 4.18852 | + | 0.108326i | 2.61631 | + | 0.801736i | −0.221150 | − | 4.30359i | 0.558840 | − | 1.39354i | −5.44433 | − | 0.211265i | −2.97525 | + | 0.384562i | −6.48293 | − | 2.07867i |
4.7 | −2.46704 | − | 0.0318966i | −0.976238 | + | 1.43072i | 4.08596 | + | 0.105673i | −4.03733 | − | 1.23719i | 2.45406 | − | 3.49851i | −1.39476 | + | 3.47801i | −5.14609 | − | 0.199692i | −1.09392 | − | 2.79345i | 9.92081 | + | 3.18098i |
4.8 | −2.38931 | − | 0.0308915i | 0.801520 | − | 1.53544i | 3.70850 | + | 0.0959110i | 1.80500 | + | 0.553120i | −1.96251 | + | 3.64387i | 1.52892 | − | 3.81257i | −4.08235 | − | 0.158414i | −1.71513 | − | 2.46137i | −4.29561 | − | 1.37733i |
4.9 | −2.38009 | − | 0.0307723i | −1.09995 | + | 1.33795i | 3.66453 | + | 0.0947738i | 2.99225 | + | 0.916937i | 2.65915 | − | 3.15058i | −0.120994 | + | 0.301715i | −3.96199 | − | 0.153743i | −0.580208 | − | 2.94336i | −7.09359 | − | 2.27447i |
4.10 | −2.29202 | − | 0.0296337i | −1.03658 | − | 1.38762i | 3.25315 | + | 0.0841346i | 3.65994 | + | 1.12154i | 2.33475 | + | 3.21117i | −0.556114 | + | 1.38674i | −2.87282 | − | 0.111478i | −0.850984 | + | 2.87677i | −8.35542 | − | 2.67906i |
4.11 | −2.27606 | − | 0.0294274i | −1.57284 | − | 0.725368i | 3.18025 | + | 0.0822491i | −3.48668 | − | 1.06845i | 3.55854 | + | 1.69727i | 0.216487 | − | 0.539838i | −2.68694 | − | 0.104265i | 1.94768 | + | 2.28178i | 7.90445 | + | 2.53446i |
4.12 | −2.08274 | − | 0.0269280i | −1.71559 | − | 0.238211i | 2.33777 | + | 0.0604606i | −0.511598 | − | 0.156773i | 3.56673 | + | 0.542331i | −1.27737 | + | 3.18529i | −0.704645 | − | 0.0273434i | 2.88651 | + | 0.817346i | 1.06131 | + | 0.340294i |
4.13 | −2.05837 | − | 0.0266128i | 0.611800 | + | 1.62040i | 2.23685 | + | 0.0578504i | −2.65761 | − | 0.814392i | −1.21619 | − | 3.35167i | 1.70918 | − | 4.26207i | −0.488730 | − | 0.0189650i | −2.25140 | + | 1.98272i | 5.44867 | + | 1.74705i |
4.14 | −2.03766 | − | 0.0263451i | −1.63810 | + | 0.562705i | 2.15204 | + | 0.0556572i | 0.558373 | + | 0.171106i | 3.35271 | − | 1.10345i | 0.399839 | − | 0.997050i | −0.311068 | − | 0.0120709i | 2.36673 | − | 1.84353i | −1.13327 | − | 0.363367i |
4.15 | −1.93424 | − | 0.0250080i | 1.62140 | + | 0.609150i | 1.74134 | + | 0.0450353i | 0.582227 | + | 0.178416i | −3.12095 | − | 1.21879i | −1.25911 | + | 3.13976i | 0.498860 | + | 0.0193580i | 2.25787 | + | 1.97535i | −1.12171 | − | 0.359660i |
4.16 | −1.90902 | − | 0.0246819i | 0.0146033 | − | 1.73199i | 1.64441 | + | 0.0425285i | −2.27162 | − | 0.696109i | −0.0706267 | + | 3.30604i | 0.132583 | − | 0.330612i | 0.677327 | + | 0.0262834i | −2.99957 | − | 0.0505857i | 4.31938 | + | 1.38495i |
4.17 | −1.90833 | − | 0.0246729i | 1.72970 | − | 0.0902694i | 1.64178 | + | 0.0424604i | 0.325467 | + | 0.0997352i | −3.30306 | + | 0.129587i | 0.868282 | − | 2.16517i | 0.682106 | + | 0.0264688i | 2.98370 | − | 0.312277i | −0.618636 | − | 0.198358i |
4.18 | −1.82547 | − | 0.0236016i | 1.47646 | + | 0.905578i | 1.33244 | + | 0.0344602i | 2.72027 | + | 0.833593i | −2.67385 | − | 1.68795i | 1.26149 | − | 3.14568i | 1.21698 | + | 0.0472244i | 1.35986 | + | 2.67410i | −4.94609 | − | 1.58590i |
4.19 | −1.68750 | − | 0.0218178i | −0.499964 | + | 1.65832i | 0.847837 | + | 0.0219272i | 0.0353662 | + | 0.0108375i | 0.879869 | − | 2.78751i | −1.23579 | + | 3.08161i | 1.94249 | + | 0.0753776i | −2.50007 | − | 1.65820i | −0.0594438 | − | 0.0190599i |
4.20 | −1.63282 | − | 0.0211109i | 0.989331 | − | 1.42170i | 0.666340 | + | 0.0172332i | −1.81802 | − | 0.557111i | −1.64542 | + | 2.30050i | −0.319539 | + | 0.796813i | 2.17581 | + | 0.0844315i | −1.04245 | − | 2.81306i | 2.95675 | + | 0.948044i |
See next 80 embeddings (of 12960 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
729.k | even | 243 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 729.2.k.a | ✓ | 12960 |
729.k | even | 243 | 1 | inner | 729.2.k.a | ✓ | 12960 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
729.2.k.a | ✓ | 12960 | 1.a | even | 1 | 1 | trivial |
729.2.k.a | ✓ | 12960 | 729.k | even | 243 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(729, [\chi])\).