Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [38,8,Mod(5,38)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([16]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("38.5");
S:= CuspForms(chi, 8);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 38 = 2 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 38.e (of order \(9\), degree \(6\), 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: | \(11.8706309684\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −7.51754 | + | 2.73616i | −11.6345 | + | 65.9825i | 49.0268 | − | 41.1384i | −116.114 | − | 97.4309i | −93.0759 | − | 527.860i | −468.313 | − | 811.141i | −256.000 | + | 443.405i | −2163.22 | − | 787.347i | 1139.48 | + | 414.735i |
5.2 | −7.51754 | + | 2.73616i | −7.18109 | + | 40.7260i | 49.0268 | − | 41.1384i | −24.6197 | − | 20.6584i | −57.4487 | − | 325.808i | 764.198 | + | 1323.63i | −256.000 | + | 443.405i | 448.070 | + | 163.084i | 241.604 | + | 87.9368i |
5.3 | −7.51754 | + | 2.73616i | 5.10191 | − | 28.9344i | 49.0268 | − | 41.1384i | 128.722 | + | 108.010i | 40.8153 | + | 231.475i | −158.544 | − | 274.607i | −256.000 | + | 443.405i | 1243.94 | + | 452.756i | −1263.21 | − | 459.769i |
5.4 | −7.51754 | + | 2.73616i | 7.10410 | − | 40.2894i | 49.0268 | − | 41.1384i | −241.265 | − | 202.445i | 56.8328 | + | 322.315i | −369.505 | − | 640.002i | −256.000 | + | 443.405i | 482.343 | + | 175.558i | 2367.64 | + | 861.750i |
5.5 | −7.51754 | + | 2.73616i | 11.5048 | − | 65.2472i | 49.0268 | − | 41.1384i | 158.287 | + | 132.818i | 92.0387 | + | 521.978i | 668.912 | + | 1158.59i | −256.000 | + | 443.405i | −2069.73 | − | 753.319i | −1553.34 | − | 565.369i |
9.1 | 1.38919 | + | 7.87846i | −46.9776 | + | 39.4189i | −60.1403 | + | 21.8893i | −248.833 | − | 90.5677i | −375.821 | − | 315.351i | 81.2847 | − | 140.789i | −256.000 | − | 443.405i | 273.278 | − | 1549.84i | 367.859 | − | 2086.23i |
9.2 | 1.38919 | + | 7.87846i | −21.2048 | + | 17.7929i | −60.1403 | + | 21.8893i | 194.773 | + | 70.8916i | −169.638 | − | 142.344i | −485.627 | + | 841.131i | −256.000 | − | 443.405i | −246.714 | + | 1399.18i | −287.941 | + | 1632.99i |
9.3 | 1.38919 | + | 7.87846i | −9.71662 | + | 8.15321i | −60.1403 | + | 21.8893i | 424.315 | + | 154.438i | −77.7330 | − | 65.2257i | 847.859 | − | 1468.54i | −256.000 | − | 443.405i | −351.831 | + | 1995.33i | −627.282 | + | 3557.49i |
9.4 | 1.38919 | + | 7.87846i | 28.3430 | − | 23.7826i | −60.1403 | + | 21.8893i | −54.5194 | − | 19.8434i | 226.744 | + | 190.261i | −356.201 | + | 616.958i | −256.000 | − | 443.405i | −142.054 | + | 805.630i | 80.5982 | − | 457.095i |
9.5 | 1.38919 | + | 7.87846i | 45.5653 | − | 38.2339i | −60.1403 | + | 21.8893i | −199.214 | − | 72.5079i | 364.523 | + | 305.871i | 458.368 | − | 793.917i | −256.000 | − | 443.405i | 234.604 | − | 1330.50i | 294.506 | − | 1670.23i |
17.1 | 1.38919 | − | 7.87846i | −46.9776 | − | 39.4189i | −60.1403 | − | 21.8893i | −248.833 | + | 90.5677i | −375.821 | + | 315.351i | 81.2847 | + | 140.789i | −256.000 | + | 443.405i | 273.278 | + | 1549.84i | 367.859 | + | 2086.23i |
17.2 | 1.38919 | − | 7.87846i | −21.2048 | − | 17.7929i | −60.1403 | − | 21.8893i | 194.773 | − | 70.8916i | −169.638 | + | 142.344i | −485.627 | − | 841.131i | −256.000 | + | 443.405i | −246.714 | − | 1399.18i | −287.941 | − | 1632.99i |
17.3 | 1.38919 | − | 7.87846i | −9.71662 | − | 8.15321i | −60.1403 | − | 21.8893i | 424.315 | − | 154.438i | −77.7330 | + | 65.2257i | 847.859 | + | 1468.54i | −256.000 | + | 443.405i | −351.831 | − | 1995.33i | −627.282 | − | 3557.49i |
17.4 | 1.38919 | − | 7.87846i | 28.3430 | + | 23.7826i | −60.1403 | − | 21.8893i | −54.5194 | + | 19.8434i | 226.744 | − | 190.261i | −356.201 | − | 616.958i | −256.000 | + | 443.405i | −142.054 | − | 805.630i | 80.5982 | + | 457.095i |
17.5 | 1.38919 | − | 7.87846i | 45.5653 | + | 38.2339i | −60.1403 | − | 21.8893i | −199.214 | + | 72.5079i | 364.523 | − | 305.871i | 458.368 | + | 793.917i | −256.000 | + | 443.405i | 234.604 | + | 1330.50i | 294.506 | + | 1670.23i |
23.1 | −7.51754 | − | 2.73616i | −11.6345 | − | 65.9825i | 49.0268 | + | 41.1384i | −116.114 | + | 97.4309i | −93.0759 | + | 527.860i | −468.313 | + | 811.141i | −256.000 | − | 443.405i | −2163.22 | + | 787.347i | 1139.48 | − | 414.735i |
23.2 | −7.51754 | − | 2.73616i | −7.18109 | − | 40.7260i | 49.0268 | + | 41.1384i | −24.6197 | + | 20.6584i | −57.4487 | + | 325.808i | 764.198 | − | 1323.63i | −256.000 | − | 443.405i | 448.070 | − | 163.084i | 241.604 | − | 87.9368i |
23.3 | −7.51754 | − | 2.73616i | 5.10191 | + | 28.9344i | 49.0268 | + | 41.1384i | 128.722 | − | 108.010i | 40.8153 | − | 231.475i | −158.544 | + | 274.607i | −256.000 | − | 443.405i | 1243.94 | − | 452.756i | −1263.21 | + | 459.769i |
23.4 | −7.51754 | − | 2.73616i | 7.10410 | + | 40.2894i | 49.0268 | + | 41.1384i | −241.265 | + | 202.445i | 56.8328 | − | 322.315i | −369.505 | + | 640.002i | −256.000 | − | 443.405i | 482.343 | − | 175.558i | 2367.64 | − | 861.750i |
23.5 | −7.51754 | − | 2.73616i | 11.5048 | + | 65.2472i | 49.0268 | + | 41.1384i | 158.287 | − | 132.818i | 92.0387 | − | 521.978i | 668.912 | − | 1158.59i | −256.000 | − | 443.405i | −2069.73 | + | 753.319i | −1553.34 | + | 565.369i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 38.8.e.a | ✓ | 30 |
19.e | even | 9 | 1 | inner | 38.8.e.a | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.8.e.a | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
38.8.e.a | ✓ | 30 | 19.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{30} - 45 T_{3}^{29} + 2565 T_{3}^{28} - 99603 T_{3}^{27} - 5241915 T_{3}^{26} + \cdots + 10\!\cdots\!61 \) acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).