Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [176,2,Mod(5,176)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(176, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 5, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("176.5");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 176 = 2^{4} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 176.w (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: | \(1.40536707557\) |
Analytic rank: | \(0\) |
Dimension: | \(176\) |
Relative dimension: | \(22\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −1.39423 | + | 0.236897i | 3.10742 | − | 0.492167i | 1.88776 | − | 0.660577i | −1.27797 | + | 0.651156i | −4.21587 | + | 1.42233i | 2.30623 | + | 3.17425i | −2.47549 | + | 1.36820i | 6.56066 | − | 2.13169i | 1.62752 | − | 1.21061i |
5.2 | −1.37854 | + | 0.315659i | 0.901291 | − | 0.142750i | 1.80072 | − | 0.870295i | 3.45027 | − | 1.75800i | −1.19740 | + | 0.481287i | −1.81367 | − | 2.49631i | −2.20764 | + | 1.76815i | −2.06122 | + | 0.669732i | −4.20139 | + | 3.51258i |
5.3 | −1.37053 | − | 0.348781i | −0.417657 | + | 0.0661504i | 1.75670 | + | 0.956029i | −0.736011 | + | 0.375016i | 0.595484 | + | 0.0550098i | 1.33653 | + | 1.83957i | −2.07417 | − | 1.92297i | −2.68311 | + | 0.871795i | 1.13952 | − | 0.257265i |
5.4 | −1.34050 | + | 0.450612i | −3.05021 | + | 0.483105i | 1.59390 | − | 1.20809i | −2.67423 | + | 1.36259i | 3.87112 | − | 2.02206i | −0.413323 | − | 0.568890i | −1.59224 | + | 2.33768i | 6.21720 | − | 2.02009i | 2.97081 | − | 3.03159i |
5.5 | −1.17638 | − | 0.784944i | −1.93030 | + | 0.305729i | 0.767727 | + | 1.84678i | 1.24626 | − | 0.635003i | 2.51074 | + | 1.15552i | −1.09820 | − | 1.51155i | 0.546481 | − | 2.77513i | 0.779411 | − | 0.253246i | −1.96452 | − | 0.231243i |
5.6 | −1.02618 | − | 0.973113i | 2.68580 | − | 0.425388i | 0.106100 | + | 1.99718i | 0.102082 | − | 0.0520132i | −3.17007 | − | 2.17706i | −2.36506 | − | 3.25523i | 1.83461 | − | 2.15272i | 4.17938 | − | 1.35796i | −0.155369 | − | 0.0459620i |
5.7 | −0.779948 | + | 1.17970i | 1.33374 | − | 0.211244i | −0.783362 | − | 1.84020i | 0.141837 | − | 0.0722696i | −0.791046 | + | 1.73817i | 1.45574 | + | 2.00365i | 2.78186 | + | 0.511133i | −1.11893 | + | 0.363561i | −0.0253694 | + | 0.223691i |
5.8 | −0.592330 | + | 1.28419i | −2.54744 | + | 0.403475i | −1.29829 | − | 1.52133i | 2.98748 | − | 1.52220i | 0.990787 | − | 3.51039i | −1.07945 | − | 1.48574i | 2.72269 | − | 0.766123i | 3.47349 | − | 1.12860i | 0.185217 | + | 4.73814i |
5.9 | −0.565804 | − | 1.29610i | 0.310032 | − | 0.0491042i | −1.35973 | + | 1.46667i | −3.81933 | + | 1.94605i | −0.239061 | − | 0.374048i | 1.09214 | + | 1.50320i | 2.67029 | + | 0.932494i | −2.75946 | + | 0.896603i | 4.68326 | + | 3.84914i |
5.10 | −0.520421 | − | 1.31498i | 1.22266 | − | 0.193650i | −1.45832 | + | 1.36868i | 2.51383 | − | 1.28086i | −0.890940 | − | 1.50698i | 2.24956 | + | 3.09626i | 2.55873 | + | 1.20537i | −1.39578 | + | 0.453517i | −2.99255 | − | 2.63904i |
5.11 | −0.287066 | + | 1.38477i | −0.925602 | + | 0.146601i | −1.83519 | − | 0.795041i | −1.96526 | + | 1.00135i | 0.0626996 | − | 1.32383i | 0.432076 | + | 0.594702i | 1.62777 | − | 2.31309i | −2.01792 | + | 0.655663i | −0.822483 | − | 3.00889i |
5.12 | 0.0503613 | + | 1.41332i | 2.39469 | − | 0.379282i | −1.99493 | + | 0.142353i | 2.15797 | − | 1.09954i | 0.656645 | + | 3.36536i | −0.678287 | − | 0.933582i | −0.301657 | − | 2.81230i | 2.73753 | − | 0.889476i | 1.66268 | + | 2.99452i |
5.13 | 0.0542058 | − | 1.41317i | −3.21042 | + | 0.508480i | −1.99412 | − | 0.153205i | 0.266677 | − | 0.135879i | 0.544548 | + | 4.56444i | 1.68453 | + | 2.31856i | −0.324598 | + | 2.80974i | 7.19505 | − | 2.33782i | −0.177565 | − | 0.384227i |
5.14 | 0.165330 | − | 1.40452i | −0.757844 | + | 0.120031i | −1.94533 | − | 0.464417i | −0.814347 | + | 0.414930i | 0.0432907 | + | 1.08425i | −2.91240 | − | 4.00857i | −0.973904 | + | 2.65547i | −2.29325 | + | 0.745122i | 0.448140 | + | 1.21236i |
5.15 | 0.664023 | + | 1.24863i | −1.67818 | + | 0.265797i | −1.11815 | + | 1.65824i | −1.78176 | + | 0.907854i | −1.44623 | − | 1.91892i | −2.03350 | − | 2.79887i | −2.81300 | − | 0.295040i | −0.107544 | + | 0.0349431i | −2.31670 | − | 1.62192i |
5.16 | 0.760498 | + | 1.19233i | −1.17460 | + | 0.186039i | −0.843285 | + | 1.81352i | 2.66815 | − | 1.35949i | −1.11510 | − | 1.25903i | 2.54995 | + | 3.50970i | −2.80363 | + | 0.373710i | −1.50808 | + | 0.490006i | 3.65008 | + | 2.14742i |
5.17 | 0.818556 | − | 1.15324i | 0.562832 | − | 0.0891439i | −0.659933 | − | 1.88799i | 1.83674 | − | 0.935866i | 0.357905 | − | 0.722051i | 0.677755 | + | 0.932849i | −2.71750 | − | 0.784359i | −2.54434 | + | 0.826705i | 0.424195 | − | 2.88426i |
5.18 | 0.994957 | + | 1.00502i | 2.09875 | − | 0.332409i | −0.0201206 | + | 1.99990i | −1.30316 | + | 0.663991i | 2.42224 | + | 1.77855i | −0.801284 | − | 1.10287i | −2.02995 | + | 1.96959i | 1.44108 | − | 0.468236i | −1.96391 | − | 0.649052i |
5.19 | 1.11969 | − | 0.863883i | 2.35196 | − | 0.372513i | 0.507411 | − | 1.93456i | −3.24121 | + | 1.65148i | 2.31165 | − | 2.44892i | 0.349519 | + | 0.481071i | −1.10309 | − | 2.60445i | 2.53977 | − | 0.825220i | −2.20247 | + | 4.64917i |
5.20 | 1.29627 | − | 0.565399i | −2.20058 | + | 0.348538i | 1.36065 | − | 1.46582i | 2.16222 | − | 1.10171i | −2.65549 | + | 1.69601i | −0.296642 | − | 0.408292i | 0.934999 | − | 2.66942i | 1.86792 | − | 0.606923i | 2.17993 | − | 2.65063i |
See next 80 embeddings (of 176 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
16.e | even | 4 | 1 | inner |
176.w | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 176.2.w.a | ✓ | 176 |
4.b | odd | 2 | 1 | 704.2.be.a | 176 | ||
11.c | even | 5 | 1 | inner | 176.2.w.a | ✓ | 176 |
16.e | even | 4 | 1 | inner | 176.2.w.a | ✓ | 176 |
16.f | odd | 4 | 1 | 704.2.be.a | 176 | ||
44.h | odd | 10 | 1 | 704.2.be.a | 176 | ||
176.v | odd | 20 | 1 | 704.2.be.a | 176 | ||
176.w | even | 20 | 1 | inner | 176.2.w.a | ✓ | 176 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
176.2.w.a | ✓ | 176 | 1.a | even | 1 | 1 | trivial |
176.2.w.a | ✓ | 176 | 11.c | even | 5 | 1 | inner |
176.2.w.a | ✓ | 176 | 16.e | even | 4 | 1 | inner |
176.2.w.a | ✓ | 176 | 176.w | even | 20 | 1 | inner |
704.2.be.a | 176 | 4.b | odd | 2 | 1 | ||
704.2.be.a | 176 | 16.f | odd | 4 | 1 | ||
704.2.be.a | 176 | 44.h | odd | 10 | 1 | ||
704.2.be.a | 176 | 176.v | odd | 20 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(176, [\chi])\).