Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [864,2,Mod(11,864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(72))
chi = DirichletCharacter(H, H._module([36, 45, 52]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("864.11");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 864 = 2^{5} \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 864.bt (of order \(72\), degree \(24\), 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: | \(3408\) |
Relative dimension: | \(142\) over \(\Q(\zeta_{72})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{72}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −1.41414 | − | 0.0139384i | 0.185106 | − | 1.72213i | 1.99961 | + | 0.0394217i | 3.08870 | + | 2.83028i | −0.285770 | + | 2.43276i | 1.76195 | + | 2.51633i | −2.82719 | − | 0.0836194i | −2.93147 | − | 0.637552i | −4.32843 | − | 4.04547i |
11.2 | −1.41363 | + | 0.0407906i | 0.204734 | − | 1.71991i | 1.99667 | − | 0.115325i | −1.35883 | − | 1.24514i | −0.219261 | + | 2.43966i | 1.85804 | + | 2.65356i | −2.81784 | + | 0.244472i | −2.91617 | − | 0.704247i | 1.97166 | + | 1.70473i |
11.3 | −1.41271 | + | 0.0652429i | −1.34471 | + | 1.09168i | 1.99149 | − | 0.184338i | 1.03243 | + | 0.946048i | 1.82845 | − | 1.62996i | −1.95935 | − | 2.79824i | −2.80136 | + | 0.390347i | 0.616467 | − | 2.93598i | −1.52025 | − | 1.26913i |
11.4 | −1.41165 | + | 0.0851035i | 1.59667 | + | 0.671291i | 1.98551 | − | 0.240273i | 2.40984 | + | 2.20821i | −2.31108 | − | 0.811745i | 0.164322 | + | 0.234676i | −2.78241 | + | 0.508155i | 2.09874 | + | 2.14366i | −3.58978 | − | 2.91214i |
11.5 | −1.41018 | + | 0.106692i | −0.243550 | + | 1.71484i | 1.97723 | − | 0.300912i | −0.0643323 | − | 0.0589497i | 0.160489 | − | 2.44423i | 2.92052 | + | 4.17093i | −2.75616 | + | 0.635297i | −2.88137 | − | 0.835299i | 0.0970098 | + | 0.0762661i |
11.6 | −1.40429 | − | 0.167206i | −0.631256 | − | 1.61292i | 1.94408 | + | 0.469613i | −2.68720 | − | 2.46236i | 0.616779 | + | 2.37057i | −2.07740 | − | 2.96684i | −2.65154 | − | 0.984538i | −2.20303 | + | 2.03633i | 3.36189 | + | 3.90719i |
11.7 | −1.40277 | + | 0.179519i | 1.46202 | − | 0.928701i | 1.93555 | − | 0.503650i | 0.162424 | + | 0.148834i | −1.88417 | + | 1.56522i | 0.0602653 | + | 0.0860678i | −2.62472 | + | 1.05398i | 1.27503 | − | 2.71557i | −0.254563 | − | 0.179622i |
11.8 | −1.39800 | + | 0.213530i | −1.67838 | − | 0.427824i | 1.90881 | − | 0.597031i | −0.187368 | − | 0.171692i | 2.43773 | + | 0.239713i | −0.568924 | − | 0.812508i | −2.54103 | + | 1.24224i | 2.63393 | + | 1.43611i | 0.298602 | + | 0.200016i |
11.9 | −1.39297 | + | 0.244177i | −0.927439 | + | 1.46282i | 1.88075 | − | 0.680266i | −2.98399 | − | 2.73432i | 0.934710 | − | 2.26414i | −0.441286 | − | 0.630221i | −2.45374 | + | 1.40683i | −1.27971 | − | 2.71336i | 4.82428 | + | 3.08022i |
11.10 | −1.38679 | − | 0.277148i | 0.934968 | + | 1.45802i | 1.84638 | + | 0.768693i | −1.28907 | − | 1.18122i | −0.892516 | − | 2.28110i | −0.239361 | − | 0.341842i | −2.34750 | − | 1.57774i | −1.25167 | + | 2.72641i | 1.46030 | + | 1.99537i |
11.11 | −1.38529 | − | 0.284573i | 0.873142 | − | 1.49587i | 1.83804 | + | 0.788430i | −0.201543 | − | 0.184680i | −1.63524 | + | 1.82373i | −1.85768 | − | 2.65304i | −2.32184 | − | 1.61526i | −1.47525 | − | 2.61221i | 0.226640 | + | 0.313189i |
11.12 | −1.36394 | + | 0.373712i | −1.63885 | + | 0.560498i | 1.72068 | − | 1.01944i | 1.13561 | + | 1.04059i | 2.02584 | − | 1.37695i | 2.47485 | + | 3.53445i | −1.96593 | + | 2.03350i | 2.37168 | − | 1.83715i | −1.93778 | − | 0.994916i |
11.13 | −1.35632 | − | 0.400507i | −1.37398 | − | 1.05460i | 1.67919 | + | 1.08643i | 0.982807 | + | 0.900577i | 1.44117 | + | 1.98066i | 0.399771 | + | 0.570933i | −1.84239 | − | 2.14607i | 0.775631 | + | 2.89800i | −0.972311 | − | 1.61509i |
11.14 | −1.35346 | − | 0.410054i | −0.153915 | + | 1.72520i | 1.66371 | + | 1.10998i | 2.53456 | + | 2.32250i | 0.915743 | − | 2.27187i | 0.366857 | + | 0.523926i | −1.79661 | − | 2.18453i | −2.95262 | − | 0.531068i | −2.47808 | − | 4.18272i |
11.15 | −1.35292 | − | 0.411830i | 1.69193 | + | 0.370633i | 1.66079 | + | 1.11435i | −2.86206 | − | 2.62259i | −2.13641 | − | 1.19823i | 2.22100 | + | 3.17192i | −1.78800 | − | 2.19159i | 2.72526 | + | 1.25417i | 2.79207 | + | 4.72684i |
11.16 | −1.33659 | + | 0.462092i | −1.08282 | − | 1.35185i | 1.57294 | − | 1.23525i | 1.59945 | + | 1.46563i | 2.07196 | + | 1.30651i | −1.32812 | − | 1.89675i | −1.53158 | + | 2.37787i | −0.655021 | + | 2.92762i | −2.81507 | − | 1.21985i |
11.17 | −1.32493 | + | 0.494525i | 1.65521 | + | 0.510185i | 1.51089 | − | 1.31042i | −1.77522 | − | 1.62669i | −2.44534 | + | 0.142581i | −2.52489 | − | 3.60592i | −1.35379 | + | 2.48340i | 2.47942 | + | 1.68892i | 3.15648 | + | 1.27736i |
11.18 | −1.31349 | − | 0.524166i | −1.41051 | + | 1.00522i | 1.45050 | + | 1.37697i | 1.01410 | + | 0.929248i | 2.37959 | − | 0.581007i | −0.562323 | − | 0.803081i | −1.18345 | − | 2.56894i | 0.979057 | − | 2.83574i | −0.844922 | − | 1.75211i |
11.19 | −1.30373 | − | 0.547978i | 1.71908 | + | 0.211552i | 1.39944 | + | 1.42883i | 0.540899 | + | 0.495643i | −2.12530 | − | 1.21783i | −1.09679 | − | 1.56638i | −1.04153 | − | 2.62968i | 2.91049 | + | 0.727352i | −0.433587 | − | 0.942587i |
11.20 | −1.29919 | + | 0.558675i | 1.58138 | − | 0.706576i | 1.37577 | − | 1.45164i | −2.22832 | − | 2.04188i | −1.65975 | + | 1.80145i | 1.01305 | + | 1.44679i | −0.976377 | + | 2.65456i | 2.00150 | − | 2.23473i | 4.03574 | + | 1.40787i |
See next 80 embeddings (of 3408 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.f | odd | 18 | 1 | inner |
32.h | odd | 8 | 1 | inner |
864.bt | even | 72 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 864.2.bt.a | ✓ | 3408 |
27.f | odd | 18 | 1 | inner | 864.2.bt.a | ✓ | 3408 |
32.h | odd | 8 | 1 | inner | 864.2.bt.a | ✓ | 3408 |
864.bt | even | 72 | 1 | inner | 864.2.bt.a | ✓ | 3408 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
864.2.bt.a | ✓ | 3408 | 1.a | even | 1 | 1 | trivial |
864.2.bt.a | ✓ | 3408 | 27.f | odd | 18 | 1 | inner |
864.2.bt.a | ✓ | 3408 | 32.h | odd | 8 | 1 | inner |
864.2.bt.a | ✓ | 3408 | 864.bt | even | 72 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(864, [\chi])\).