Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,3,Mod(53,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 5, 6]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 160.bb (of order \(8\), degree \(4\), 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: | \(4.35968422976\) |
Analytic rank: | \(0\) |
Dimension: | \(184\) |
Relative dimension: | \(46\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
53.1 | −1.99895 | − | 0.0647219i | 2.83311 | + | 1.17351i | 3.99162 | + | 0.258752i | −3.94886 | − | 3.06700i | −5.58730 | − | 2.52916i | −10.8175 | −7.96232 | − | 0.775578i | 0.285413 | + | 0.285413i | 7.69509 | + | 6.38637i | ||
53.2 | −1.99869 | + | 0.0723062i | −0.937700 | − | 0.388408i | 3.98954 | − | 0.289036i | 3.72867 | + | 3.33122i | 1.90226 | + | 0.708507i | 2.36359 | −7.95297 | + | 0.866163i | −5.63554 | − | 5.63554i | −7.69333 | − | 6.38848i | ||
53.3 | −1.97081 | + | 0.340438i | −2.80621 | − | 1.16237i | 3.76820 | − | 1.34188i | 1.18740 | − | 4.85696i | 5.92623 | + | 1.33547i | 1.22791 | −6.96959 | + | 3.92744i | 0.159755 | + | 0.159755i | −0.686648 | + | 9.97640i | ||
53.4 | −1.95783 | + | 0.408524i | −4.62398 | − | 1.91532i | 3.66622 | − | 1.59964i | −3.33317 | + | 3.72692i | 9.83544 | + | 1.86086i | −10.0343 | −6.52435 | + | 4.62957i | 11.3488 | + | 11.3488i | 5.00326 | − | 8.65837i | ||
53.5 | −1.94738 | + | 0.455756i | 5.04734 | + | 2.09068i | 3.58457 | − | 1.77506i | 1.86459 | + | 4.63932i | −10.7819 | − | 1.77098i | 5.36005 | −6.17153 | + | 5.09041i | 14.7407 | + | 14.7407i | −5.74546 | − | 8.18472i | ||
53.6 | −1.87059 | − | 0.707744i | −1.23386 | − | 0.511082i | 2.99820 | + | 2.64779i | −4.18639 | + | 2.73389i | 1.94633 | + | 1.82928i | 11.1763 | −3.73443 | − | 7.07489i | −5.10275 | − | 5.10275i | 9.76590 | − | 2.15109i | ||
53.7 | −1.81433 | − | 0.841556i | 2.98308 | + | 1.23563i | 2.58357 | + | 3.05372i | 4.00144 | − | 2.99808i | −4.37243 | − | 4.75227i | 3.22829 | −2.11756 | − | 7.71466i | 1.00802 | + | 1.00802i | −9.78297 | + | 2.07205i | ||
53.8 | −1.75049 | + | 0.967352i | 1.84758 | + | 0.765291i | 2.12846 | − | 3.38669i | −4.83879 | − | 1.25939i | −3.97448 | + | 0.447618i | 8.29433 | −0.449741 | + | 7.98735i | −3.53610 | − | 3.53610i | 9.68856 | − | 2.47625i | ||
53.9 | −1.60685 | − | 1.19081i | −5.49113 | − | 2.27450i | 1.16395 | + | 3.82691i | 4.99245 | − | 0.274630i | 6.11495 | + | 10.1937i | 6.10452 | 2.68682 | − | 7.53532i | 18.6152 | + | 18.6152i | −8.34917 | − | 5.50376i | ||
53.10 | −1.54645 | − | 1.26826i | −0.233667 | − | 0.0967881i | 0.783021 | + | 3.92261i | 3.06987 | + | 3.94663i | 0.238602 | + | 0.446029i | −11.9595 | 3.76400 | − | 7.05920i | −6.31873 | − | 6.31873i | 0.257954 | − | 9.99667i | ||
53.11 | −1.52127 | + | 1.29836i | −0.105501 | − | 0.0436998i | 0.628542 | − | 3.95031i | 4.49805 | − | 2.18348i | 0.217233 | − | 0.0704980i | −3.47322 | 4.17272 | + | 6.82557i | −6.35474 | − | 6.35474i | −4.00782 | + | 9.16174i | ||
53.12 | −1.46084 | − | 1.36599i | −2.98434 | − | 1.23615i | 0.268122 | + | 3.99100i | −4.69474 | − | 1.72030i | 2.67107 | + | 5.88241i | −5.77525 | 5.06000 | − | 6.19648i | 1.01424 | + | 1.01424i | 4.50836 | + | 8.92607i | ||
53.13 | −1.39655 | + | 1.43166i | 1.12933 | + | 0.467783i | −0.0993084 | − | 3.99877i | −1.30374 | + | 4.82703i | −2.24687 | + | 0.963533i | −5.91992 | 5.86357 | + | 5.44229i | −5.30740 | − | 5.30740i | −5.08994 | − | 8.60770i | ||
53.14 | −1.28127 | − | 1.53569i | 4.44428 | + | 1.84088i | −0.716715 | + | 3.93527i | −3.55622 | + | 3.51473i | −2.86727 | − | 9.18371i | −1.60224 | 6.96167 | − | 3.94147i | 9.99881 | + | 9.99881i | 9.95401 | + | 0.957970i | ||
53.15 | −1.22944 | + | 1.57749i | −3.88606 | − | 1.60966i | −0.976973 | − | 3.87886i | −3.87919 | − | 3.15466i | 7.31689 | − | 4.15126i | 6.72772 | 7.32000 | + | 3.22764i | 6.14650 | + | 6.14650i | 9.74567 | − | 2.24095i | ||
53.16 | −1.11009 | + | 1.66364i | 4.97343 | + | 2.06006i | −1.53540 | − | 3.69358i | 1.42035 | − | 4.79402i | −8.94815 | + | 5.98714i | −2.45065 | 7.84922 | + | 1.54586i | 14.1272 | + | 14.1272i | 6.39881 | + | 7.68474i | ||
53.17 | −0.963660 | − | 1.75253i | 1.59304 | + | 0.659857i | −2.14272 | + | 3.37768i | −2.41012 | − | 4.38079i | −0.378725 | − | 3.42772i | 7.82512 | 7.98434 | + | 0.500244i | −4.26161 | − | 4.26161i | −5.35493 | + | 8.44540i | ||
53.18 | −0.628589 | − | 1.89865i | −1.59431 | − | 0.660386i | −3.20975 | + | 2.38694i | 3.34875 | − | 3.71293i | −0.251675 | + | 3.44216i | −6.24134 | 6.54958 | + | 4.59379i | −4.25824 | − | 4.25824i | −9.15455 | − | 4.02421i | ||
53.19 | −0.575434 | + | 1.91543i | −4.38228 | − | 1.81520i | −3.33775 | − | 2.20441i | 4.90866 | − | 0.951362i | 5.99861 | − | 7.34943i | −8.87308 | 6.14305 | − | 5.12474i | 9.54550 | + | 9.54550i | −1.00234 | + | 9.94964i | ||
53.20 | −0.483550 | − | 1.94066i | −2.23592 | − | 0.926148i | −3.53236 | + | 1.87682i | 0.525403 | + | 4.97232i | −0.716165 | + | 4.78701i | 7.42339 | 5.35034 | + | 5.94759i | −2.22238 | − | 2.22238i | 9.39554 | − | 3.42399i | ||
See next 80 embeddings (of 184 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
160.bb | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 160.3.bb.a | yes | 184 |
5.c | odd | 4 | 1 | 160.3.v.a | ✓ | 184 | |
32.g | even | 8 | 1 | 160.3.v.a | ✓ | 184 | |
160.bb | odd | 8 | 1 | inner | 160.3.bb.a | yes | 184 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
160.3.v.a | ✓ | 184 | 5.c | odd | 4 | 1 | |
160.3.v.a | ✓ | 184 | 32.g | even | 8 | 1 | |
160.3.bb.a | yes | 184 | 1.a | even | 1 | 1 | trivial |
160.3.bb.a | yes | 184 | 160.bb | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(160, [\chi])\).