Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [241,4,Mod(6,241)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(241, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([11]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("241.6");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 241 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 241.l (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: | \(14.2194603114\) |
Analytic rank: | \(0\) |
Dimension: | \(480\) |
Relative dimension: | \(60\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6.1 | − | 5.43744i | −8.71742 | − | 2.83246i | −21.5657 | 8.02439 | − | 2.60728i | −15.4013 | + | 47.4004i | 0.140911 | − | 0.889677i | 73.7627i | 46.1271 | + | 33.5133i | −14.1769 | − | 43.6321i | |||||
6.2 | − | 5.39315i | 3.26132 | + | 1.05967i | −21.0861 | 3.81423 | − | 1.23932i | 5.71494 | − | 17.5888i | 4.44762 | − | 28.0812i | 70.5750i | −12.3302 | − | 8.95838i | −6.68382 | − | 20.5707i | |||||
6.3 | − | 5.24721i | 8.70976 | + | 2.82997i | −19.5332 | 16.8575 | − | 5.47733i | 14.8495 | − | 45.7019i | −5.00286 | + | 31.5868i | 60.5171i | 46.0078 | + | 33.4266i | −28.7407 | − | 88.4547i | |||||
6.4 | − | 5.12174i | −1.39929 | − | 0.454658i | −18.2323 | −1.79124 | + | 0.582010i | −2.32864 | + | 7.16682i | −3.59462 | + | 22.6955i | 52.4070i | −20.0921 | − | 14.5978i | 2.98091 | + | 9.17429i | |||||
6.5 | − | 5.10495i | −3.74663 | − | 1.21736i | −18.0605 | −18.6948 | + | 6.07432i | −6.21454 | + | 19.1264i | 1.63685 | − | 10.3347i | 51.3585i | −9.28815 | − | 6.74823i | 31.0091 | + | 95.4363i | |||||
6.6 | − | 4.88246i | 7.17944 | + | 2.33274i | −15.8384 | −12.9862 | + | 4.21946i | 11.3895 | − | 35.0534i | −0.200118 | + | 1.26350i | 38.2709i | 24.2593 | + | 17.6254i | 20.6014 | + | 63.4045i | |||||
6.7 | − | 4.79602i | −2.45247 | − | 0.796855i | −15.0018 | 18.7999 | − | 6.10844i | −3.82173 | + | 11.7621i | −0.959050 | + | 6.05520i | 33.5806i | −16.4638 | − | 11.9617i | −29.2962 | − | 90.1644i | |||||
6.8 | − | 4.38477i | 5.62014 | + | 1.82610i | −11.2262 | 5.05605 | − | 1.64281i | 8.00700 | − | 24.6430i | 2.16461 | − | 13.6668i | 14.1461i | 6.40792 | + | 4.65563i | −7.20334 | − | 22.1696i | |||||
6.9 | − | 4.00444i | −8.72794 | − | 2.83588i | −8.03551 | −9.94745 | + | 3.23212i | −11.3561 | + | 34.9505i | −1.84489 | + | 11.6482i | 0.142209i | 46.2912 | + | 33.6326i | 12.9428 | + | 39.8339i | |||||
6.10 | − | 4.00161i | −5.16474 | − | 1.67813i | −8.01286 | 7.05912 | − | 2.29365i | −6.71520 | + | 20.6673i | 4.28106 | − | 27.0295i | 0.0514781i | 2.01497 | + | 1.46396i | −9.17828 | − | 28.2478i | |||||
6.11 | − | 3.91882i | 0.586333 | + | 0.190511i | −7.35714 | −0.392752 | + | 0.127613i | 0.746578 | − | 2.29773i | −1.26854 | + | 8.00924i | − | 2.51926i | −21.5360 | − | 15.6468i | 0.500091 | + | 1.53912i | ||||
6.12 | − | 3.79472i | −5.87698 | − | 1.90955i | −6.39992 | 2.07643 | − | 0.674674i | −7.24620 | + | 22.3015i | 3.48733 | − | 22.0181i | − | 6.07186i | 9.04911 | + | 6.57456i | −2.56020 | − | 7.87949i | ||||
6.13 | − | 3.68420i | 6.98766 | + | 2.27043i | −5.57332 | −17.1608 | + | 5.57587i | 8.36471 | − | 25.7439i | −1.27701 | + | 8.06271i | − | 8.94036i | 21.8291 | + | 15.8598i | 20.5426 | + | 63.2237i | ||||
6.14 | − | 3.54891i | 3.03932 | + | 0.987536i | −4.59474 | −3.89545 | + | 1.26571i | 3.50467 | − | 10.7863i | −4.50462 | + | 28.4411i | − | 12.0850i | −13.5812 | − | 9.86732i | 4.49188 | + | 13.8246i | ||||
6.15 | − | 3.25737i | −1.57299 | − | 0.511095i | −2.61046 | −12.4034 | + | 4.03011i | −1.66483 | + | 5.12381i | 2.48904 | − | 15.7152i | − | 17.5557i | −19.6304 | − | 14.2623i | 13.1275 | + | 40.4024i | ||||
6.16 | − | 3.24273i | −6.30302 | − | 2.04798i | −2.51533 | 15.8065 | − | 5.13585i | −6.64104 | + | 20.4390i | −3.45395 | + | 21.8074i | − | 17.7853i | 13.6904 | + | 9.94665i | −16.6542 | − | 51.2564i | ||||
6.17 | − | 3.07869i | 4.02143 | + | 1.30664i | −1.47831 | 16.8403 | − | 5.47173i | 4.02274 | − | 12.3807i | 0.828297 | − | 5.22966i | − | 20.0782i | −7.37887 | − | 5.36106i | −16.8458 | − | 51.8459i | ||||
6.18 | − | 2.80316i | 9.26647 | + | 3.01086i | 0.142309 | 6.15659 | − | 2.00040i | 8.43991 | − | 25.9754i | 2.85184 | − | 18.0058i | − | 22.8242i | 54.9588 | + | 39.9299i | −5.60743 | − | 17.2579i | ||||
6.19 | − | 2.00769i | −6.37017 | − | 2.06979i | 3.96917 | 6.20933 | − | 2.01753i | −4.15551 | + | 12.7893i | −2.94910 | + | 18.6199i | − | 24.0304i | 14.4515 | + | 10.4996i | −4.05059 | − | 12.4664i | ||||
6.20 | − | 1.99525i | 4.72936 | + | 1.53666i | 4.01896 | −11.2852 | + | 3.66677i | 3.06603 | − | 9.43628i | 4.97729 | − | 31.4253i | − | 23.9809i | −1.83794 | − | 1.33534i | 7.31614 | + | 22.5168i | ||||
See next 80 embeddings (of 480 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
241.l | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 241.4.l.a | ✓ | 480 |
241.l | even | 20 | 1 | inner | 241.4.l.a | ✓ | 480 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
241.4.l.a | ✓ | 480 | 1.a | even | 1 | 1 | trivial |
241.4.l.a | ✓ | 480 | 241.l | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(241, [\chi])\).