Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(8,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([6, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.by (of order \(20\), degree \(8\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −1.27226 | − | 2.49695i | 0.229619 | − | 0.316044i | −3.44055 | + | 4.73551i | −0.373022 | − | 0.513421i | −1.08128 | − | 0.171258i | −0.158002 | + | 0.997588i | 10.6658 | + | 1.68930i | 0.879892 | + | 2.70803i | −0.807406 | + | 1.58462i |
8.2 | −1.19854 | − | 2.35228i | 2.01676 | − | 2.77583i | −2.92112 | + | 4.02058i | −1.84630 | − | 2.54121i | −8.94670 | − | 1.41702i | 0.360188 | − | 2.27414i | 7.74356 | + | 1.22646i | −2.71087 | − | 8.34321i | −3.76476 | + | 7.38875i |
8.3 | −1.15648 | − | 2.26971i | −0.804281 | + | 1.10700i | −2.63859 | + | 3.63171i | 2.20178 | + | 3.03050i | 3.44270 | + | 0.545270i | −0.640182 | + | 4.04195i | 6.26242 | + | 0.991869i | 0.348475 | + | 1.07250i | 4.33205 | − | 8.50212i |
8.4 | −1.15551 | − | 2.26781i | −0.590903 | + | 0.813308i | −2.63221 | + | 3.62292i | −0.341463 | − | 0.469983i | 2.52722 | + | 0.400273i | 0.492045 | − | 3.10665i | 6.22986 | + | 0.986713i | 0.614748 | + | 1.89200i | −0.671271 | + | 1.31744i |
8.5 | −1.14775 | − | 2.25259i | −1.61160 | + | 2.21818i | −2.58126 | + | 3.55279i | 1.58808 | + | 2.18581i | 6.84636 | + | 1.08436i | 0.552167 | − | 3.48625i | 5.97160 | + | 0.945809i | −1.39600 | − | 4.29645i | 3.10101 | − | 6.08607i |
8.6 | −1.10242 | − | 2.16362i | 0.888199 | − | 1.22250i | −2.29036 | + | 3.15242i | 0.977174 | + | 1.34496i | −3.62420 | − | 0.574017i | 0.241995 | − | 1.52790i | 4.54880 | + | 0.720459i | 0.221439 | + | 0.681519i | 1.83274 | − | 3.59695i |
8.7 | −1.09577 | − | 2.15058i | 0.591002 | − | 0.813444i | −2.24869 | + | 3.09506i | −1.43415 | − | 1.97393i | −2.39698 | − | 0.379644i | −0.471977 | + | 2.97995i | 4.35236 | + | 0.689347i | 0.614643 | + | 1.89168i | −2.67360 | + | 5.24723i |
8.8 | −1.08112 | − | 2.12183i | −1.44320 | + | 1.98639i | −2.15774 | + | 2.96988i | −2.06979 | − | 2.84882i | 5.77505 | + | 0.914677i | 0.0942794 | − | 0.595256i | 3.93023 | + | 0.622487i | −0.935876 | − | 2.88033i | −3.80700 | + | 7.47165i |
8.9 | −0.993007 | − | 1.94889i | 1.27965 | − | 1.76129i | −1.63652 | + | 2.25248i | 1.37971 | + | 1.89901i | −4.70325 | − | 0.744922i | −0.548105 | + | 3.46060i | 1.69419 | + | 0.268333i | −0.537581 | − | 1.65450i | 2.33089 | − | 4.57463i |
8.10 | −0.951186 | − | 1.86681i | 1.65015 | − | 2.27123i | −1.40464 | + | 1.93333i | 0.779238 | + | 1.07253i | −5.80955 | − | 0.920142i | −0.134991 | + | 0.852298i | 0.806480 | + | 0.127734i | −1.50846 | − | 4.64257i | 1.26100 | − | 2.47486i |
8.11 | −0.873559 | − | 1.71446i | −1.30288 | + | 1.79327i | −1.00068 | + | 1.37732i | −0.684503 | − | 0.942138i | 4.21263 | + | 0.667214i | −0.105751 | + | 0.667685i | −0.565458 | − | 0.0895598i | −0.591247 | − | 1.81967i | −1.01730 | + | 1.99656i |
8.12 | −0.857178 | − | 1.68231i | −0.511539 | + | 0.704073i | −0.919830 | + | 1.26604i | 0.525408 | + | 0.723162i | 1.62295 | + | 0.257049i | 0.291102 | − | 1.83795i | −0.811382 | − | 0.128510i | 0.693004 | + | 2.13285i | 0.766212 | − | 1.50378i |
8.13 | −0.835532 | − | 1.63982i | −0.262985 | + | 0.361967i | −0.815339 | + | 1.12222i | −1.83486 | − | 2.52547i | 0.813295 | + | 0.128813i | −0.270525 | + | 1.70803i | −1.11404 | − | 0.176446i | 0.865192 | + | 2.66279i | −2.60824 | + | 5.11897i |
8.14 | −0.791651 | − | 1.55370i | 0.665663 | − | 0.916207i | −0.611709 | + | 0.841946i | −1.41987 | − | 1.95429i | −1.95049 | − | 0.308927i | 0.725085 | − | 4.57800i | −1.65219 | − | 0.261681i | 0.530724 | + | 1.63340i | −1.91234 | + | 3.75318i |
8.15 | −0.775075 | − | 1.52117i | 0.0969981 | − | 0.133506i | −0.537648 | + | 0.740009i | 1.78404 | + | 2.45552i | −0.278267 | − | 0.0440732i | 0.397079 | − | 2.50706i | −1.83006 | − | 0.289854i | 0.918636 | + | 2.82727i | 2.35250 | − | 4.61704i |
8.16 | −0.774310 | − | 1.51967i | −1.70097 | + | 2.34118i | −0.534269 | + | 0.735359i | −0.0702390 | − | 0.0966757i | 4.87490 | + | 0.772109i | −0.473066 | + | 2.98682i | −1.83794 | − | 0.291101i | −1.66079 | − | 5.11139i | −0.0925284 | + | 0.181597i |
8.17 | −0.663369 | − | 1.30194i | 1.51607 | − | 2.08669i | −0.0794070 | + | 0.109294i | 0.218773 | + | 0.301116i | −3.72246 | − | 0.589579i | 0.376420 | − | 2.37662i | −2.69144 | − | 0.426283i | −1.12877 | − | 3.47399i | 0.246906 | − | 0.484580i |
8.18 | −0.616127 | − | 1.20922i | −1.63169 | + | 2.24582i | 0.0929757 | − | 0.127970i | 1.58212 | + | 2.17760i | 3.72101 | + | 0.589351i | −0.301788 | + | 1.90541i | −2.89288 | − | 0.458188i | −1.45427 | − | 4.47578i | 1.65840 | − | 3.25480i |
8.19 | −0.555563 | − | 1.09035i | 1.02315 | − | 1.40825i | 0.295348 | − | 0.406511i | −1.71747 | − | 2.36389i | −2.10392 | − | 0.333228i | 0.0499679 | − | 0.315485i | −3.02466 | − | 0.479059i | −0.00927322 | − | 0.0285400i | −1.62332 | + | 3.18594i |
8.20 | −0.514453 | − | 1.00967i | 0.385562 | − | 0.530680i | 0.420797 | − | 0.579178i | 0.416499 | + | 0.573262i | −0.734165 | − | 0.116280i | −0.694842 | + | 4.38706i | −3.03972 | − | 0.481444i | 0.794087 | + | 2.44395i | 0.364536 | − | 0.715443i |
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
671.by | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.by.a | yes | 480 |
11.d | odd | 10 | 1 | 671.2.bt.a | ✓ | 480 | |
61.j | odd | 20 | 1 | 671.2.bt.a | ✓ | 480 | |
671.by | even | 20 | 1 | inner | 671.2.by.a | yes | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.bt.a | ✓ | 480 | 11.d | odd | 10 | 1 | |
671.2.bt.a | ✓ | 480 | 61.j | odd | 20 | 1 | |
671.2.by.a | yes | 480 | 1.a | even | 1 | 1 | trivial |
671.2.by.a | yes | 480 | 671.by | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).