Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,6,Mod(2,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.2");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 33.f (of order \(10\), 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: | \(5.29266605383\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{10})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −3.08173 | + | 9.48458i | −12.5347 | + | 9.26724i | −54.5716 | − | 39.6486i | 16.3931 | − | 5.32643i | −49.2674 | − | 147.445i | −74.1699 | + | 102.086i | 286.047 | − | 207.825i | 71.2363 | − | 232.324i | 171.896i | ||
2.2 | −2.95379 | + | 9.09083i | 10.1153 | − | 11.8609i | −48.0298 | − | 34.8957i | 91.6682 | − | 29.7848i | 77.9468 | + | 126.991i | 70.5766 | − | 97.1403i | 211.641 | − | 153.766i | −38.3609 | − | 239.953i | 921.319i | ||
2.3 | −2.81967 | + | 8.67805i | 12.9564 | + | 8.66789i | −41.4695 | − | 30.1293i | −53.2208 | + | 17.2925i | −111.753 | + | 87.9955i | −14.1075 | + | 19.4173i | 142.170 | − | 103.293i | 92.7354 | + | 224.609i | − | 510.612i | |
2.4 | −2.32473 | + | 7.15479i | −12.7649 | − | 8.94746i | −19.8981 | − | 14.4568i | −28.0945 | + | 9.12845i | 93.6922 | − | 70.5298i | 146.291 | − | 201.352i | −45.0663 | + | 32.7426i | 82.8858 | + | 228.427i | − | 222.231i | |
2.5 | −1.97194 | + | 6.06902i | 4.72506 | − | 14.8551i | −7.05585 | − | 5.12637i | −59.6930 | + | 19.3954i | 80.8382 | + | 57.9699i | −97.0552 | + | 133.585i | −120.178 | + | 87.3143i | −198.348 | − | 140.382i | − | 400.525i | |
2.6 | −1.36557 | + | 4.20281i | 6.31381 | + | 14.2526i | 10.0898 | + | 7.33064i | 98.1154 | − | 31.8796i | −68.5228 | + | 7.07275i | −5.71539 | + | 7.86655i | −158.991 | + | 115.514i | −163.272 | + | 179.976i | 455.894i | ||
2.7 | −1.12695 | + | 3.46841i | −12.9075 | − | 8.74056i | 15.1287 | + | 10.9916i | 54.0225 | − | 17.5530i | 44.8620 | − | 34.9182i | −123.528 | + | 170.022i | −149.586 | + | 108.681i | 90.2051 | + | 225.637i | 207.153i | ||
2.8 | −1.01916 | + | 3.13665i | −8.08821 | + | 13.3259i | 17.0887 | + | 12.4156i | −40.8503 | + | 13.2731i | −33.5556 | − | 38.9511i | 39.2343 | − | 54.0014i | −141.742 | + | 102.981i | −112.162 | − | 215.566i | − | 141.660i | |
2.9 | −0.489986 | + | 1.50802i | 15.5884 | + | 0.0191829i | 23.8545 | + | 17.3313i | −16.6412 | + | 5.40705i | −7.66704 | + | 23.4983i | 36.5406 | − | 50.2938i | −78.8739 | + | 57.3052i | 242.999 | + | 0.598063i | − | 27.7446i | |
2.10 | 0.489986 | − | 1.50802i | 4.79885 | − | 14.8314i | 23.8545 | + | 17.3313i | 16.6412 | − | 5.40705i | −20.0147 | − | 14.5039i | 36.5406 | − | 50.2938i | 78.8739 | − | 57.3052i | −196.942 | − | 142.348i | − | 27.7446i | |
2.11 | 1.01916 | − | 3.13665i | −15.1731 | + | 3.57441i | 17.0887 | + | 12.4156i | 40.8503 | − | 13.2731i | −4.25217 | + | 51.2356i | 39.2343 | − | 54.0014i | 141.742 | − | 102.981i | 217.447 | − | 108.470i | − | 141.660i | |
2.12 | 1.12695 | − | 3.46841i | 4.32414 | + | 14.9767i | 15.1287 | + | 10.9916i | −54.0225 | + | 17.5530i | 56.8185 | + | 1.88017i | −123.528 | + | 170.022i | 149.586 | − | 108.681i | −205.604 | + | 129.523i | 207.153i | ||
2.13 | 1.36557 | − | 4.20281i | −11.6039 | − | 10.4091i | 10.0898 | + | 7.33064i | −98.1154 | + | 31.8796i | −59.5934 | + | 34.5547i | −5.71539 | + | 7.86655i | 158.991 | − | 115.514i | 26.3023 | + | 241.572i | 455.894i | ||
2.14 | 1.97194 | − | 6.06902i | 15.5882 | + | 0.0966769i | −7.05585 | − | 5.12637i | 59.6930 | − | 19.3954i | 31.3257 | − | 94.4141i | −97.0552 | + | 133.585i | 120.178 | − | 87.3143i | 242.981 | + | 3.01403i | − | 400.525i | |
2.15 | 2.32473 | − | 7.15479i | 4.56497 | + | 14.9051i | −19.8981 | − | 14.4568i | 28.0945 | − | 9.12845i | 117.255 | + | 1.98889i | 146.291 | − | 201.352i | 45.0663 | − | 32.7426i | −201.322 | + | 136.082i | − | 222.231i | |
2.16 | 2.81967 | − | 8.67805i | −4.23991 | − | 15.0008i | −41.4695 | − | 30.1293i | 53.2208 | − | 17.2925i | −142.133 | − | 5.50305i | −14.1075 | + | 19.4173i | −142.170 | + | 103.293i | −207.046 | + | 127.204i | − | 510.612i | |
2.17 | 2.95379 | − | 9.09083i | 14.4062 | − | 5.95502i | −48.0298 | − | 34.8957i | −91.6682 | + | 29.7848i | −11.5833 | − | 148.554i | 70.5766 | − | 97.1403i | −211.641 | + | 153.766i | 172.075 | − | 171.578i | 921.319i | ||
2.18 | 3.08173 | − | 9.48458i | −12.6871 | + | 9.05745i | −54.5716 | − | 39.6486i | −16.3931 | + | 5.32643i | 46.8079 | + | 148.244i | −74.1699 | + | 102.086i | −286.047 | + | 207.825i | 78.9251 | − | 229.826i | 171.896i | ||
8.1 | −8.92830 | + | 6.48679i | −5.92302 | + | 14.4194i | 27.7475 | − | 85.3981i | 33.7792 | − | 46.4931i | −40.6528 | − | 167.162i | −11.4606 | − | 3.72378i | 197.091 | + | 606.584i | −172.836 | − | 170.812i | 634.222i | ||
8.2 | −7.80447 | + | 5.67028i | 14.8651 | − | 4.69358i | 18.8692 | − | 58.0733i | −46.3038 | + | 63.7318i | −89.4001 | + | 120.920i | −135.347 | − | 43.9769i | 86.6348 | + | 266.635i | 198.941 | − | 139.541i | − | 759.949i | |
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
11.d | odd | 10 | 1 | inner |
33.f | even | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 33.6.f.a | ✓ | 72 |
3.b | odd | 2 | 1 | inner | 33.6.f.a | ✓ | 72 |
11.d | odd | 10 | 1 | inner | 33.6.f.a | ✓ | 72 |
33.f | even | 10 | 1 | inner | 33.6.f.a | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.6.f.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
33.6.f.a | ✓ | 72 | 3.b | odd | 2 | 1 | inner |
33.6.f.a | ✓ | 72 | 11.d | odd | 10 | 1 | inner |
33.6.f.a | ✓ | 72 | 33.f | even | 10 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(33, [\chi])\).