Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [720,2,Mod(197,720)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(720, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("720.197");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 720.bc (of order \(4\), degree \(2\), 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: | \(5.74922894553\) |
Analytic rank: | \(0\) |
Dimension: | \(96\) |
Relative dimension: | \(48\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
197.1 | −1.41326 | + | 0.0520118i | 0 | 1.99459 | − | 0.147012i | 2.14437 | − | 0.633793i | 0 | 0.431588 | − | 0.431588i | −2.81122 | + | 0.311508i | 0 | −2.99758 | + | 1.00725i | ||||||
197.2 | −1.41043 | − | 0.103327i | 0 | 1.97865 | + | 0.291473i | −2.21174 | − | 0.328918i | 0 | −2.33595 | + | 2.33595i | −2.76063 | − | 0.615551i | 0 | 3.08553 | + | 0.692451i | ||||||
197.3 | −1.40624 | − | 0.149965i | 0 | 1.95502 | + | 0.421775i | −1.29201 | − | 1.82503i | 0 | −2.02860 | + | 2.02860i | −2.68598 | − | 0.886302i | 0 | 1.54318 | + | 2.76018i | ||||||
197.4 | −1.39508 | + | 0.231832i | 0 | 1.89251 | − | 0.646849i | 0.244063 | + | 2.22271i | 0 | 0.718424 | − | 0.718424i | −2.49024 | + | 1.34115i | 0 | −0.855782 | − | 3.04428i | ||||||
197.5 | −1.39415 | + | 0.237394i | 0 | 1.88729 | − | 0.661925i | 0.257710 | − | 2.22117i | 0 | 3.41086 | − | 3.41086i | −2.47402 | + | 1.37085i | 0 | 0.168007 | + | 3.15781i | ||||||
197.6 | −1.34579 | − | 0.434580i | 0 | 1.62228 | + | 1.16970i | −1.74165 | + | 1.40237i | 0 | 2.18248 | − | 2.18248i | −1.67491 | − | 2.27918i | 0 | 2.95334 | − | 1.13040i | ||||||
197.7 | −1.33026 | − | 0.480016i | 0 | 1.53917 | + | 1.27709i | 2.06437 | + | 0.859295i | 0 | −1.97431 | + | 1.97431i | −1.43447 | − | 2.43768i | 0 | −2.33366 | − | 2.13401i | ||||||
197.8 | −1.20197 | + | 0.745171i | 0 | 0.889441 | − | 1.79134i | −0.470363 | + | 2.18604i | 0 | −0.177389 | + | 0.177389i | 0.265776 | + | 2.81591i | 0 | −1.06361 | − | 2.97804i | ||||||
197.9 | −1.17483 | + | 0.787263i | 0 | 0.760434 | − | 1.84979i | −2.23567 | − | 0.0422885i | 0 | 1.80532 | − | 1.80532i | 0.562896 | + | 2.77185i | 0 | 2.65981 | − | 1.71038i | ||||||
197.10 | −1.16790 | − | 0.797498i | 0 | 0.727993 | + | 1.86280i | −2.02484 | + | 0.948701i | 0 | 1.06871 | − | 1.06871i | 0.635356 | − | 2.75614i | 0 | 3.12140 | + | 0.506814i | ||||||
197.11 | −1.13002 | + | 0.850328i | 0 | 0.553884 | − | 1.92177i | 0.885098 | − | 2.05344i | 0 | −2.95475 | + | 2.95475i | 1.00824 | + | 2.64262i | 0 | 0.745918 | + | 3.07305i | ||||||
197.12 | −1.03923 | − | 0.959172i | 0 | 0.159980 | + | 1.99359i | 0.117317 | − | 2.23299i | 0 | −0.303980 | + | 0.303980i | 1.74594 | − | 2.22524i | 0 | −2.26374 | + | 2.20805i | ||||||
197.13 | −1.02160 | − | 0.977925i | 0 | 0.0873238 | + | 1.99809i | 1.73447 | − | 1.41125i | 0 | −2.32146 | + | 2.32146i | 1.86478 | − | 2.12664i | 0 | −3.15202 | − | 0.254450i | ||||||
197.14 | −0.981505 | + | 1.01816i | 0 | −0.0732970 | − | 1.99866i | 1.78433 | + | 1.34765i | 0 | −3.52140 | + | 3.52140i | 2.10689 | + | 1.88706i | 0 | −3.12345 | + | 0.494009i | ||||||
197.15 | −0.910521 | − | 1.08211i | 0 | −0.341905 | + | 1.97056i | 2.04823 | + | 0.897076i | 0 | 3.15125 | − | 3.15125i | 2.44366 | − | 1.42426i | 0 | −0.894226 | − | 3.03321i | ||||||
197.16 | −0.898132 | + | 1.09241i | 0 | −0.386719 | − | 1.96226i | 1.29671 | − | 1.82168i | 0 | 2.16011 | − | 2.16011i | 2.49091 | + | 1.33991i | 0 | 0.825405 | + | 3.05266i | ||||||
197.17 | −0.663127 | + | 1.24910i | 0 | −1.12053 | − | 1.65663i | −1.95867 | − | 1.07871i | 0 | 1.03362 | − | 1.03362i | 2.81235 | − | 0.301100i | 0 | 2.64626 | − | 1.73127i | ||||||
197.18 | −0.572467 | − | 1.29317i | 0 | −1.34456 | + | 1.48059i | −0.0482170 | − | 2.23555i | 0 | 2.70126 | − | 2.70126i | 2.68437 | + | 0.891154i | 0 | −2.86333 | + | 1.34213i | ||||||
197.19 | −0.487044 | − | 1.32770i | 0 | −1.52558 | + | 1.29330i | 1.08017 | + | 1.95786i | 0 | 0.362166 | − | 0.362166i | 2.46013 | + | 1.39562i | 0 | 2.07336 | − | 2.38771i | ||||||
197.20 | −0.427613 | + | 1.34802i | 0 | −1.63429 | − | 1.15286i | −1.09255 | − | 1.95099i | 0 | −1.70979 | + | 1.70979i | 2.25292 | − | 1.71008i | 0 | 3.09715 | − | 0.638502i | ||||||
See all 96 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
80.i | odd | 4 | 1 | inner |
240.bb | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 720.2.bc.a | ✓ | 96 |
3.b | odd | 2 | 1 | inner | 720.2.bc.a | ✓ | 96 |
4.b | odd | 2 | 1 | 2880.2.bc.a | 96 | ||
5.c | odd | 4 | 1 | 720.2.bg.a | yes | 96 | |
12.b | even | 2 | 1 | 2880.2.bc.a | 96 | ||
15.e | even | 4 | 1 | 720.2.bg.a | yes | 96 | |
16.e | even | 4 | 1 | 720.2.bg.a | yes | 96 | |
16.f | odd | 4 | 1 | 2880.2.bg.a | 96 | ||
20.e | even | 4 | 1 | 2880.2.bg.a | 96 | ||
48.i | odd | 4 | 1 | 720.2.bg.a | yes | 96 | |
48.k | even | 4 | 1 | 2880.2.bg.a | 96 | ||
60.l | odd | 4 | 1 | 2880.2.bg.a | 96 | ||
80.i | odd | 4 | 1 | inner | 720.2.bc.a | ✓ | 96 |
80.s | even | 4 | 1 | 2880.2.bc.a | 96 | ||
240.z | odd | 4 | 1 | 2880.2.bc.a | 96 | ||
240.bb | even | 4 | 1 | inner | 720.2.bc.a | ✓ | 96 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
720.2.bc.a | ✓ | 96 | 1.a | even | 1 | 1 | trivial |
720.2.bc.a | ✓ | 96 | 3.b | odd | 2 | 1 | inner |
720.2.bc.a | ✓ | 96 | 80.i | odd | 4 | 1 | inner |
720.2.bc.a | ✓ | 96 | 240.bb | even | 4 | 1 | inner |
720.2.bg.a | yes | 96 | 5.c | odd | 4 | 1 | |
720.2.bg.a | yes | 96 | 15.e | even | 4 | 1 | |
720.2.bg.a | yes | 96 | 16.e | even | 4 | 1 | |
720.2.bg.a | yes | 96 | 48.i | odd | 4 | 1 | |
2880.2.bc.a | 96 | 4.b | odd | 2 | 1 | ||
2880.2.bc.a | 96 | 12.b | even | 2 | 1 | ||
2880.2.bc.a | 96 | 80.s | even | 4 | 1 | ||
2880.2.bc.a | 96 | 240.z | odd | 4 | 1 | ||
2880.2.bg.a | 96 | 16.f | odd | 4 | 1 | ||
2880.2.bg.a | 96 | 20.e | even | 4 | 1 | ||
2880.2.bg.a | 96 | 48.k | even | 4 | 1 | ||
2880.2.bg.a | 96 | 60.l | odd | 4 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).