Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [980,2,Mod(43,980)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(28))
chi = DirichletCharacter(H, H._module([14, 21, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("980.43");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.bk (of order \(28\), degree \(12\), 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: | \(7.82533939809\) |
Analytic rank: | \(0\) |
Dimension: | \(1968\) |
Relative dimension: | \(164\) over \(\Q(\zeta_{28})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{28}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | −1.41226 | − | 0.0742447i | 1.98344 | − | 0.694037i | 1.98898 | + | 0.209706i | 1.71471 | − | 1.43519i | −2.85267 | + | 0.832902i | 1.76581 | − | 1.97026i | −2.79339 | − | 0.443831i | 1.10686 | − | 0.882694i | −2.52818 | + | 1.89955i |
43.2 | −1.41030 | + | 0.105184i | −1.21166 | + | 0.423978i | 1.97787 | − | 0.296680i | 2.23335 | + | 0.110254i | 1.66420 | − | 0.725381i | 0.246144 | − | 2.63428i | −2.75818 | + | 0.626446i | −1.05713 | + | 0.843035i | −3.16128 | + | 0.0794211i |
43.3 | −1.40933 | − | 0.117431i | 0.998651 | − | 0.349443i | 1.97242 | + | 0.330997i | −1.20433 | − | 1.88403i | −1.44846 | + | 0.375208i | −1.03437 | − | 2.43518i | −2.74092 | − | 0.698107i | −1.47030 | + | 1.17253i | 1.47606 | + | 2.79665i |
43.4 | −1.40577 | − | 0.154339i | 0.258629 | − | 0.0904982i | 1.95236 | + | 0.433929i | −2.21946 | − | 0.272005i | −0.377539 | + | 0.0873028i | 2.07765 | − | 1.63810i | −2.67759 | − | 0.911328i | −2.28680 | + | 1.82366i | 3.07806 | + | 0.724925i |
43.5 | −1.40555 | + | 0.156312i | 2.81238 | − | 0.984094i | 1.95113 | − | 0.439409i | −2.13943 | + | 0.650274i | −3.79911 | + | 1.82280i | 2.61319 | + | 0.413788i | −2.67373 | + | 0.922596i | 4.59555 | − | 3.66483i | 2.90542 | − | 1.24841i |
43.6 | −1.40487 | + | 0.162324i | −1.05229 | + | 0.368211i | 1.94730 | − | 0.456088i | 0.731251 | − | 2.11312i | 1.41855 | − | 0.688098i | −2.31427 | + | 1.28225i | −2.66167 | + | 0.956836i | −1.37377 | + | 1.09554i | −0.684300 | + | 3.08735i |
43.7 | −1.40060 | − | 0.195769i | −1.56249 | + | 0.546739i | 1.92335 | + | 0.548388i | 1.59083 | + | 1.57139i | 2.29546 | − | 0.459874i | −0.889736 | + | 2.49166i | −2.58648 | − | 1.14460i | −0.203042 | + | 0.161920i | −1.92048 | − | 2.51232i |
43.8 | −1.39416 | + | 0.237318i | 1.38411 | − | 0.484323i | 1.88736 | − | 0.661720i | 1.88572 | + | 1.20168i | −1.81474 | + | 1.00370i | 1.49236 | + | 2.18469i | −2.47424 | + | 1.37045i | −0.664290 | + | 0.529753i | −2.91418 | − | 1.22782i |
43.9 | −1.39351 | + | 0.241111i | −2.14193 | + | 0.749494i | 1.88373 | − | 0.671980i | −1.35940 | + | 1.77540i | 2.80409 | − | 1.56087i | 2.35594 | + | 1.20397i | −2.46297 | + | 1.39060i | 1.68063 | − | 1.34026i | 1.46626 | − | 2.80180i |
43.10 | −1.39184 | − | 0.250568i | −0.428666 | + | 0.149997i | 1.87443 | + | 0.697500i | −0.746109 | − | 2.10792i | 0.634219 | − | 0.101361i | 1.68855 | + | 2.03686i | −2.43414 | − | 1.44048i | −2.18424 | + | 1.74187i | 0.510288 | + | 3.12083i |
43.11 | −1.38528 | − | 0.284609i | −0.414553 | + | 0.145058i | 1.83800 | + | 0.788526i | −1.55889 | + | 1.60308i | 0.615556 | − | 0.0829608i | −1.96396 | − | 1.77281i | −2.32171 | − | 1.61544i | −2.19468 | + | 1.75020i | 2.61574 | − | 1.77704i |
43.12 | −1.36731 | + | 0.361193i | −2.75717 | + | 0.964776i | 1.73908 | − | 0.987726i | −0.559737 | − | 2.16488i | 3.42144 | − | 2.31502i | 2.51037 | − | 0.835501i | −2.02110 | + | 1.97867i | 4.32571 | − | 3.44964i | 1.54727 | + | 2.75789i |
43.13 | −1.36674 | − | 0.363362i | 2.20437 | − | 0.771341i | 1.73594 | + | 0.993239i | −1.41697 | + | 1.72979i | −3.29306 | + | 0.253238i | −0.756704 | + | 2.53523i | −2.01166 | − | 1.98827i | 1.91877 | − | 1.53016i | 2.56517 | − | 1.84930i |
43.14 | −1.36143 | + | 0.382773i | 3.08329 | − | 1.07889i | 1.70697 | − | 1.04224i | 1.91380 | − | 1.15645i | −3.78471 | + | 2.64903i | −1.66528 | + | 2.05593i | −1.92498 | + | 2.07231i | 5.99720 | − | 4.78261i | −2.16284 | + | 2.30698i |
43.15 | −1.35823 | − | 0.393966i | −3.16048 | + | 1.10590i | 1.68958 | + | 1.07019i | 0.255185 | + | 2.22146i | 4.72835 | − | 0.256947i | 0.216393 | − | 2.63689i | −1.87322 | − | 2.11921i | 6.42013 | − | 5.11988i | 0.528578 | − | 3.11779i |
43.16 | −1.35733 | + | 0.397058i | 1.33271 | − | 0.466334i | 1.68469 | − | 1.07788i | 1.41300 | + | 1.73304i | −1.62376 | + | 1.16213i | −2.51513 | − | 0.821039i | −1.85870 | + | 2.13195i | −0.786858 | + | 0.627498i | −2.60603 | − | 1.79126i |
43.17 | −1.35462 | + | 0.406221i | 0.871746 | − | 0.305037i | 1.66997 | − | 1.10055i | −2.06822 | − | 0.849992i | −1.05697 | + | 0.767330i | −2.43982 | + | 1.02337i | −1.81510 | + | 2.16920i | −1.67860 | + | 1.33864i | 3.14692 | + | 0.311259i |
43.18 | −1.34319 | − | 0.442547i | 2.84445 | − | 0.995317i | 1.60830 | + | 1.18885i | 0.866001 | + | 2.06156i | −4.26111 | + | 0.0780940i | −1.64391 | − | 2.07306i | −1.63413 | − | 2.30859i | 4.75476 | − | 3.79179i | −0.250864 | − | 3.15231i |
43.19 | −1.33432 | + | 0.468610i | −2.46734 | + | 0.863358i | 1.56081 | − | 1.25055i | −2.23559 | − | 0.0463024i | 2.88763 | − | 2.30821i | −2.62139 | + | 0.358211i | −1.49659 | + | 2.40004i | 2.99686 | − | 2.38992i | 3.00468 | − | 0.985838i |
43.20 | −1.33030 | − | 0.479898i | 1.61150 | − | 0.563887i | 1.53940 | + | 1.27682i | 1.35364 | − | 1.77979i | −2.41438 | − | 0.0232149i | −2.64196 | + | 0.141657i | −1.43512 | − | 2.43730i | −0.0665419 | + | 0.0530654i | −2.65486 | + | 1.71805i |
See next 80 embeddings (of 1968 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
20.e | even | 4 | 1 | inner |
49.e | even | 7 | 1 | inner |
196.k | odd | 14 | 1 | inner |
245.r | odd | 28 | 1 | inner |
980.bk | even | 28 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.2.bk.a | ✓ | 1968 |
4.b | odd | 2 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
5.c | odd | 4 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
20.e | even | 4 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
49.e | even | 7 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
196.k | odd | 14 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
245.r | odd | 28 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
980.bk | even | 28 | 1 | inner | 980.2.bk.a | ✓ | 1968 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
980.2.bk.a | ✓ | 1968 | 1.a | even | 1 | 1 | trivial |
980.2.bk.a | ✓ | 1968 | 4.b | odd | 2 | 1 | inner |
980.2.bk.a | ✓ | 1968 | 5.c | odd | 4 | 1 | inner |
980.2.bk.a | ✓ | 1968 | 20.e | even | 4 | 1 | inner |
980.2.bk.a | ✓ | 1968 | 49.e | even | 7 | 1 | inner |
980.2.bk.a | ✓ | 1968 | 196.k | odd | 14 | 1 | inner |
980.2.bk.a | ✓ | 1968 | 245.r | odd | 28 | 1 | inner |
980.2.bk.a | ✓ | 1968 | 980.bk | even | 28 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(980, [\chi])\).