Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,8,Mod(37,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.37");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.f (of order \(5\), 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: | \(30.9261175229\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(7\) over \(\Q(\zeta_{5})\) |
Twist minimal: | no (minimal twist has level 33) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 | −16.9629 | − | 12.3242i | 0 | 96.2976 | + | 296.374i | 380.493 | − | 276.444i | 0 | 39.7826 | + | 122.438i | 1189.75 | − | 3661.68i | 0 | −9861.22 | ||||||||
37.2 | −14.6841 | − | 10.6686i | 0 | 62.2484 | + | 191.581i | −425.412 | + | 309.080i | 0 | −184.854 | − | 568.921i | 411.912 | − | 1267.74i | 0 | 9544.22 | ||||||||
37.3 | −7.82727 | − | 5.68684i | 0 | −10.6283 | − | 32.7104i | 100.934 | − | 73.3327i | 0 | −177.127 | − | 545.140i | −485.517 | + | 1494.27i | 0 | −1207.07 | ||||||||
37.4 | 3.24396 | + | 2.35687i | 0 | −34.5858 | − | 106.444i | −354.050 | + | 257.233i | 0 | 116.729 | + | 359.256i | 297.283 | − | 914.942i | 0 | −1754.79 | ||||||||
37.5 | 3.93160 | + | 2.85647i | 0 | −32.2561 | − | 99.2742i | 221.480 | − | 160.915i | 0 | −146.721 | − | 451.561i | 348.978 | − | 1074.04i | 0 | 1330.42 | ||||||||
37.6 | 12.1293 | + | 8.81245i | 0 | 29.9064 | + | 92.0423i | 3.10808 | − | 2.25815i | 0 | 300.631 | + | 925.246i | 144.646 | − | 445.175i | 0 | 57.5987 | ||||||||
37.7 | 17.1972 | + | 12.4945i | 0 | 100.077 | + | 308.005i | −190.798 | + | 138.623i | 0 | −503.289 | − | 1548.96i | −1286.53 | + | 3959.52i | 0 | −5013.22 | ||||||||
64.1 | −4.95371 | + | 15.2460i | 0 | −104.346 | − | 75.8118i | −99.6953 | − | 306.831i | 0 | 803.319 | + | 583.645i | 12.6937 | − | 9.22251i | 0 | 5171.79 | ||||||||
64.2 | −2.75376 | + | 8.47521i | 0 | 39.3082 | + | 28.5591i | 42.2440 | + | 130.014i | 0 | 29.1631 | + | 21.1883i | −1273.10 | + | 924.960i | 0 | −1218.22 | ||||||||
64.3 | −0.813458 | + | 2.50357i | 0 | 97.9481 | + | 71.1634i | −78.7070 | − | 242.235i | 0 | −1135.08 | − | 824.681i | −530.435 | + | 385.384i | 0 | 670.476 | ||||||||
64.4 | 0.970201 | − | 2.98597i | 0 | 95.5794 | + | 69.4425i | 100.399 | + | 308.998i | 0 | 1340.18 | + | 973.701i | 625.207 | − | 454.240i | 0 | 1020.07 | ||||||||
64.5 | 2.99223 | − | 9.20914i | 0 | 27.6994 | + | 20.1248i | 60.2993 | + | 185.582i | 0 | −765.654 | − | 556.280i | 1270.94 | − | 923.389i | 0 | 1889.48 | ||||||||
64.6 | 4.06039 | − | 12.4966i | 0 | −36.1242 | − | 26.2458i | −158.380 | − | 487.442i | 0 | 846.952 | + | 615.347i | 886.010 | − | 643.724i | 0 | −6734.46 | ||||||||
64.7 | 6.47025 | − | 19.9134i | 0 | −251.124 | − | 182.452i | 11.5844 | + | 35.6531i | 0 | 80.4609 | + | 58.4583i | −3089.84 | + | 2244.90i | 0 | 784.928 | ||||||||
82.1 | −4.95371 | − | 15.2460i | 0 | −104.346 | + | 75.8118i | −99.6953 | + | 306.831i | 0 | 803.319 | − | 583.645i | 12.6937 | + | 9.22251i | 0 | 5171.79 | ||||||||
82.2 | −2.75376 | − | 8.47521i | 0 | 39.3082 | − | 28.5591i | 42.2440 | − | 130.014i | 0 | 29.1631 | − | 21.1883i | −1273.10 | − | 924.960i | 0 | −1218.22 | ||||||||
82.3 | −0.813458 | − | 2.50357i | 0 | 97.9481 | − | 71.1634i | −78.7070 | + | 242.235i | 0 | −1135.08 | + | 824.681i | −530.435 | − | 385.384i | 0 | 670.476 | ||||||||
82.4 | 0.970201 | + | 2.98597i | 0 | 95.5794 | − | 69.4425i | 100.399 | − | 308.998i | 0 | 1340.18 | − | 973.701i | 625.207 | + | 454.240i | 0 | 1020.07 | ||||||||
82.5 | 2.99223 | + | 9.20914i | 0 | 27.6994 | − | 20.1248i | 60.2993 | − | 185.582i | 0 | −765.654 | + | 556.280i | 1270.94 | + | 923.389i | 0 | 1889.48 | ||||||||
82.6 | 4.06039 | + | 12.4966i | 0 | −36.1242 | + | 26.2458i | −158.380 | + | 487.442i | 0 | 846.952 | − | 615.347i | 886.010 | + | 643.724i | 0 | −6734.46 | ||||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.8.f.b | 28 | |
3.b | odd | 2 | 1 | 33.8.e.b | ✓ | 28 | |
11.c | even | 5 | 1 | inner | 99.8.f.b | 28 | |
33.f | even | 10 | 1 | 363.8.a.r | 14 | ||
33.h | odd | 10 | 1 | 33.8.e.b | ✓ | 28 | |
33.h | odd | 10 | 1 | 363.8.a.o | 14 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.8.e.b | ✓ | 28 | 3.b | odd | 2 | 1 | |
33.8.e.b | ✓ | 28 | 33.h | odd | 10 | 1 | |
99.8.f.b | 28 | 1.a | even | 1 | 1 | trivial | |
99.8.f.b | 28 | 11.c | even | 5 | 1 | inner | |
363.8.a.o | 14 | 33.h | odd | 10 | 1 | ||
363.8.a.r | 14 | 33.f | even | 10 | 1 |