Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,3,Mod(13,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 19]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.13");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 75.k (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: | \(2.04360198270\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(10\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −3.44703 | + | 0.545956i | −0.786335 | + | 1.54327i | 7.77971 | − | 2.52778i | −0.831800 | + | 4.93033i | 1.86796 | − | 5.74899i | −0.992761 | + | 0.992761i | −12.9984 | + | 6.62301i | −1.76336 | − | 2.42705i | 0.175498 | − | 17.4491i |
13.2 | −2.60745 | + | 0.412980i | 0.786335 | − | 1.54327i | 2.82402 | − | 0.917581i | −1.42490 | + | 4.79267i | −1.41299 | + | 4.34874i | 5.98338 | − | 5.98338i | 2.42430 | − | 1.23524i | −1.76336 | − | 2.42705i | 1.73607 | − | 13.0851i |
13.3 | −1.75107 | + | 0.277342i | −0.786335 | + | 1.54327i | −0.814895 | + | 0.264776i | −2.37217 | − | 4.40146i | 0.948914 | − | 2.92046i | 5.49856 | − | 5.49856i | 7.67216 | − | 3.90916i | −1.76336 | − | 2.42705i | 5.37454 | + | 7.04936i |
13.4 | −1.20062 | + | 0.190159i | 0.786335 | − | 1.54327i | −2.39891 | + | 0.779452i | −4.66031 | − | 1.81149i | −0.650620 | + | 2.00240i | −8.35473 | + | 8.35473i | 7.06431 | − | 3.59945i | −1.76336 | − | 2.42705i | 5.93972 | + | 1.28870i |
13.5 | −0.247004 | + | 0.0391215i | 0.786335 | − | 1.54327i | −3.74475 | + | 1.21674i | 3.63583 | − | 3.43231i | −0.133852 | + | 0.411956i | 8.31728 | − | 8.31728i | 1.76867 | − | 0.901180i | −1.76336 | − | 2.42705i | −0.763787 | + | 0.990032i |
13.6 | 0.367346 | − | 0.0581819i | −0.786335 | + | 1.54327i | −3.67267 | + | 1.19332i | −3.52059 | + | 3.55042i | −0.199067 | + | 0.612664i | −1.57845 | + | 1.57845i | −2.60526 | + | 1.32745i | −1.76336 | − | 2.42705i | −1.08670 | + | 1.50906i |
13.7 | 2.47604 | − | 0.392165i | −0.786335 | + | 1.54327i | 2.17273 | − | 0.705963i | 4.85676 | + | 1.18822i | −1.34178 | + | 4.12956i | 3.46290 | − | 3.46290i | −3.83175 | + | 1.95238i | −1.76336 | − | 2.42705i | 12.4915 | + | 1.03743i |
13.8 | 2.52458 | − | 0.399854i | 0.786335 | − | 1.54327i | 2.40940 | − | 0.782861i | 2.98039 | − | 4.01464i | 1.36808 | − | 4.21053i | −7.99303 | + | 7.99303i | −3.34014 | + | 1.70189i | −1.76336 | − | 2.42705i | 5.91896 | − | 11.3270i |
13.9 | 2.92729 | − | 0.463638i | 0.786335 | − | 1.54327i | 4.54985 | − | 1.47834i | −2.61452 | + | 4.26196i | 1.58631 | − | 4.88217i | 2.66748 | − | 2.66748i | 2.07035 | − | 1.05489i | −1.76336 | − | 2.42705i | −5.67746 | + | 13.6882i |
13.10 | 3.75152 | − | 0.594183i | −0.786335 | + | 1.54327i | 9.91663 | − | 3.22211i | −4.60913 | − | 1.93801i | −2.03297 | + | 6.25683i | −5.76987 | + | 5.76987i | 21.7507 | − | 11.0826i | −1.76336 | − | 2.42705i | −18.4428 | − | 4.53182i |
22.1 | −3.41817 | + | 1.74164i | −1.71073 | + | 0.270952i | 6.29941 | − | 8.67040i | 4.97139 | + | 0.534117i | 5.37565 | − | 3.90564i | −0.908106 | − | 0.908106i | −4.03119 | + | 25.4519i | 2.85317 | − | 0.927051i | −17.9233 | + | 6.83269i |
22.2 | −3.18220 | + | 1.62141i | 1.71073 | − | 0.270952i | 5.14628 | − | 7.08325i | −4.97907 | + | 0.457044i | −5.00455 | + | 3.63602i | −6.93420 | − | 6.93420i | −2.65683 | + | 16.7746i | 2.85317 | − | 0.927051i | 15.1033 | − | 9.52752i |
22.3 | −2.16137 | + | 1.10127i | 1.71073 | − | 0.270952i | 1.10757 | − | 1.52445i | 2.52133 | − | 4.31774i | −3.39912 | + | 2.46960i | 6.20191 | + | 6.20191i | 0.802843 | − | 5.06895i | 2.85317 | − | 0.927051i | −0.694505 | + | 12.1089i |
22.4 | −1.53414 | + | 0.781681i | −1.71073 | + | 0.270952i | −0.608595 | + | 0.837659i | 0.173747 | − | 4.99698i | 2.41269 | − | 1.75292i | −1.91431 | − | 1.91431i | 1.35628 | − | 8.56322i | 2.85317 | − | 0.927051i | 3.63949 | + | 7.80186i |
22.5 | −0.186256 | + | 0.0949024i | 1.71073 | − | 0.270952i | −2.32546 | + | 3.20072i | −1.31868 | + | 4.82297i | −0.292920 | + | 0.212819i | 5.11319 | + | 5.11319i | 0.260180 | − | 1.64271i | 2.85317 | − | 0.927051i | −0.212098 | − | 1.02346i |
22.6 | 0.250128 | − | 0.127447i | −1.71073 | + | 0.270952i | −2.30482 | + | 3.17231i | −4.47793 | + | 2.22444i | −0.393368 | + | 0.285799i | −6.72381 | − | 6.72381i | −0.347860 | + | 2.19630i | 2.85317 | − | 0.927051i | −0.836558 | + | 1.12709i |
22.7 | 0.510204 | − | 0.259962i | −1.71073 | + | 0.270952i | −2.15841 | + | 2.97080i | 4.69871 | + | 1.70942i | −0.802382 | + | 0.582965i | 6.06980 | + | 6.06980i | −0.687243 | + | 4.33908i | 2.85317 | − | 0.927051i | 2.84168 | − | 0.349330i |
22.8 | 1.53105 | − | 0.780110i | 1.71073 | − | 0.270952i | −0.615594 | + | 0.847292i | 4.34778 | − | 2.46917i | 2.40784 | − | 1.74940i | −3.26334 | − | 3.26334i | −1.35675 | + | 8.56621i | 2.85317 | − | 0.927051i | 4.73046 | − | 7.17217i |
22.9 | 2.73870 | − | 1.39544i | 1.71073 | − | 0.270952i | 3.20210 | − | 4.40731i | −4.71895 | + | 1.65274i | 4.30707 | − | 3.12927i | −0.250082 | − | 0.250082i | 0.696119 | − | 4.39513i | 2.85317 | − | 0.927051i | −10.6175 | + | 11.1114i |
22.10 | 2.93190 | − | 1.49388i | −1.71073 | + | 0.270952i | 4.01322 | − | 5.52372i | −0.890939 | − | 4.91998i | −4.61091 | + | 3.35002i | 4.34391 | + | 4.34391i | 1.45557 | − | 9.19012i | 2.85317 | − | 0.927051i | −9.96199 | − | 13.0939i |
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 75.3.k.a | ✓ | 80 |
3.b | odd | 2 | 1 | 225.3.r.b | 80 | ||
5.b | even | 2 | 1 | 375.3.k.a | 80 | ||
5.c | odd | 4 | 1 | 375.3.k.b | 80 | ||
5.c | odd | 4 | 1 | 375.3.k.c | 80 | ||
25.d | even | 5 | 1 | 375.3.k.c | 80 | ||
25.e | even | 10 | 1 | 375.3.k.b | 80 | ||
25.f | odd | 20 | 1 | inner | 75.3.k.a | ✓ | 80 |
25.f | odd | 20 | 1 | 375.3.k.a | 80 | ||
75.l | even | 20 | 1 | 225.3.r.b | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.3.k.a | ✓ | 80 | 1.a | even | 1 | 1 | trivial |
75.3.k.a | ✓ | 80 | 25.f | odd | 20 | 1 | inner |
225.3.r.b | 80 | 3.b | odd | 2 | 1 | ||
225.3.r.b | 80 | 75.l | even | 20 | 1 | ||
375.3.k.a | 80 | 5.b | even | 2 | 1 | ||
375.3.k.a | 80 | 25.f | odd | 20 | 1 | ||
375.3.k.b | 80 | 5.c | odd | 4 | 1 | ||
375.3.k.b | 80 | 25.e | even | 10 | 1 | ||
375.3.k.c | 80 | 5.c | odd | 4 | 1 | ||
375.3.k.c | 80 | 25.d | even | 5 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(75, [\chi])\).