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: | \(36\) |
Relative dimension: | \(6\) 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 | −13.9835 | + | 79.3043i | 49.0268 | − | 41.1384i | −397.617 | − | 333.640i | 111.868 | + | 634.435i | 170.178 | + | 294.757i | 256.000 | − | 443.405i | −4038.53 | − | 1469.91i | −3902.00 | − | 1420.21i |
5.2 | 7.51754 | − | 2.73616i | −11.7125 | + | 66.4251i | 49.0268 | − | 41.1384i | 272.185 | + | 228.391i | 93.7003 | + | 531.401i | −194.092 | − | 336.178i | 256.000 | − | 443.405i | −2220.00 | − | 808.013i | 2671.08 | + | 972.193i |
5.3 | 7.51754 | − | 2.73616i | −3.06967 | + | 17.4090i | 49.0268 | − | 41.1384i | −9.97721 | − | 8.37187i | 24.5574 | + | 139.272i | 364.735 | + | 631.739i | 256.000 | − | 443.405i | 1761.46 | + | 641.118i | −97.9109 | − | 35.6366i |
5.4 | 7.51754 | − | 2.73616i | −0.665685 | + | 3.77529i | 49.0268 | − | 41.1384i | 10.1638 | + | 8.52843i | 5.32548 | + | 30.2023i | −800.791 | − | 1387.01i | 256.000 | − | 443.405i | 2041.30 | + | 742.972i | 99.7418 | + | 36.3031i |
5.5 | 7.51754 | − | 2.73616i | 8.60073 | − | 48.7771i | 49.0268 | − | 41.1384i | 379.177 | + | 318.167i | −68.8058 | − | 390.217i | 612.206 | + | 1060.37i | 256.000 | − | 443.405i | −250.129 | − | 91.0394i | 3721.03 | + | 1354.35i |
5.6 | 7.51754 | − | 2.73616i | 11.6928 | − | 66.3132i | 49.0268 | − | 41.1384i | −157.410 | − | 132.083i | −93.5425 | − | 530.506i | −83.9701 | − | 145.440i | 256.000 | − | 443.405i | −2205.61 | − | 802.777i | −1544.74 | − | 562.238i |
9.1 | −1.38919 | − | 7.87846i | −66.5349 | + | 55.8294i | −60.1403 | + | 21.8893i | −312.664 | − | 113.800i | 532.279 | + | 446.635i | −817.968 | + | 1416.76i | 256.000 | + | 443.405i | 930.200 | − | 5275.43i | −462.223 | + | 2621.40i |
9.2 | −1.38919 | − | 7.87846i | −33.6098 | + | 28.2020i | −60.1403 | + | 21.8893i | 202.712 | + | 73.7810i | 268.878 | + | 225.616i | 234.915 | − | 406.885i | 256.000 | + | 443.405i | −45.5015 | + | 258.052i | 299.677 | − | 1699.55i |
9.3 | −1.38919 | − | 7.87846i | −12.0854 | + | 10.1408i | −60.1403 | + | 21.8893i | −307.410 | − | 111.888i | 96.6830 | + | 81.1267i | 444.097 | − | 769.199i | 256.000 | + | 443.405i | −336.549 | + | 1908.66i | −454.457 | + | 2577.35i |
9.4 | −1.38919 | − | 7.87846i | 16.0986 | − | 13.5083i | −60.1403 | + | 21.8893i | 358.329 | + | 130.421i | −128.789 | − | 108.067i | −386.454 | + | 669.359i | 256.000 | + | 443.405i | −303.078 | + | 1718.84i | 529.732 | − | 3004.26i |
9.5 | −1.38919 | − | 7.87846i | 37.7512 | − | 31.6770i | −60.1403 | + | 21.8893i | −192.429 | − | 70.0383i | −302.010 | − | 253.416i | −814.928 | + | 1411.50i | 256.000 | + | 443.405i | 41.9510 | − | 237.916i | −284.475 | + | 1613.34i |
9.6 | −1.38919 | − | 7.87846i | 65.8294 | − | 55.2375i | −60.1403 | + | 21.8893i | 133.061 | + | 48.4302i | −526.636 | − | 441.900i | 558.915 | − | 968.069i | 256.000 | + | 443.405i | 902.570 | − | 5118.73i | 196.709 | − | 1115.59i |
17.1 | −1.38919 | + | 7.87846i | −66.5349 | − | 55.8294i | −60.1403 | − | 21.8893i | −312.664 | + | 113.800i | 532.279 | − | 446.635i | −817.968 | − | 1416.76i | 256.000 | − | 443.405i | 930.200 | + | 5275.43i | −462.223 | − | 2621.40i |
17.2 | −1.38919 | + | 7.87846i | −33.6098 | − | 28.2020i | −60.1403 | − | 21.8893i | 202.712 | − | 73.7810i | 268.878 | − | 225.616i | 234.915 | + | 406.885i | 256.000 | − | 443.405i | −45.5015 | − | 258.052i | 299.677 | + | 1699.55i |
17.3 | −1.38919 | + | 7.87846i | −12.0854 | − | 10.1408i | −60.1403 | − | 21.8893i | −307.410 | + | 111.888i | 96.6830 | − | 81.1267i | 444.097 | + | 769.199i | 256.000 | − | 443.405i | −336.549 | − | 1908.66i | −454.457 | − | 2577.35i |
17.4 | −1.38919 | + | 7.87846i | 16.0986 | + | 13.5083i | −60.1403 | − | 21.8893i | 358.329 | − | 130.421i | −128.789 | + | 108.067i | −386.454 | − | 669.359i | 256.000 | − | 443.405i | −303.078 | − | 1718.84i | 529.732 | + | 3004.26i |
17.5 | −1.38919 | + | 7.87846i | 37.7512 | + | 31.6770i | −60.1403 | − | 21.8893i | −192.429 | + | 70.0383i | −302.010 | + | 253.416i | −814.928 | − | 1411.50i | 256.000 | − | 443.405i | 41.9510 | + | 237.916i | −284.475 | − | 1613.34i |
17.6 | −1.38919 | + | 7.87846i | 65.8294 | + | 55.2375i | −60.1403 | − | 21.8893i | 133.061 | − | 48.4302i | −526.636 | + | 441.900i | 558.915 | + | 968.069i | 256.000 | − | 443.405i | 902.570 | + | 5118.73i | 196.709 | + | 1115.59i |
23.1 | 7.51754 | + | 2.73616i | −13.9835 | − | 79.3043i | 49.0268 | + | 41.1384i | −397.617 | + | 333.640i | 111.868 | − | 634.435i | 170.178 | − | 294.757i | 256.000 | + | 443.405i | −4038.53 | + | 1469.91i | −3902.00 | + | 1420.21i |
23.2 | 7.51754 | + | 2.73616i | −11.7125 | − | 66.4251i | 49.0268 | + | 41.1384i | 272.185 | − | 228.391i | 93.7003 | − | 531.401i | −194.092 | + | 336.178i | 256.000 | + | 443.405i | −2220.00 | + | 808.013i | 2671.08 | − | 972.193i |
See all 36 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.b | ✓ | 36 |
19.e | even | 9 | 1 | inner | 38.8.e.b | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
38.8.e.b | ✓ | 36 | 1.a | even | 1 | 1 | trivial |
38.8.e.b | ✓ | 36 | 19.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{36} + 84 T_{3}^{35} + 2814 T_{3}^{34} - 14356 T_{3}^{33} + 10221195 T_{3}^{32} + 1921137504 T_{3}^{31} + 411923695970 T_{3}^{30} + 21156535535772 T_{3}^{29} + \cdots + 13\!\cdots\!49 \)
acting on \(S_{8}^{\mathrm{new}}(38, [\chi])\).