Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,6,Mod(11,37)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(37, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.11");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.e (of order \(6\), degree \(2\), 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: | \(5.93420133308\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
11.1 | −9.33962 | + | 5.39223i | 14.0993 | − | 24.4208i | 42.1523 | − | 73.0099i | −41.0060 | − | 23.6748i | 304.108i | −18.9093 | + | 32.7518i | 564.077i | −276.083 | − | 478.189i | 510.641 | ||||||
11.2 | −9.18598 | + | 5.30353i | −6.26292 | + | 10.8477i | 40.2548 | − | 69.7233i | 63.9646 | + | 36.9300i | − | 132.862i | 113.255 | − | 196.163i | 514.543i | 43.0517 | + | 74.5678i | −783.436 | |||||
11.3 | −7.60377 | + | 4.39004i | −8.72617 | + | 15.1142i | 22.5449 | − | 39.0489i | −72.9853 | − | 42.1381i | − | 153.233i | −77.9335 | + | 134.985i | 114.930i | −30.7921 | − | 53.3336i | 739.952 | |||||
11.4 | −6.08219 | + | 3.51156i | 3.15654 | − | 5.46728i | 8.66206 | − | 15.0031i | 30.9238 | + | 17.8538i | 44.3374i | −47.9780 | + | 83.1003i | − | 103.070i | 101.573 | + | 175.929i | −250.779 | |||||
11.5 | −5.33497 | + | 3.08014i | 3.71739 | − | 6.43870i | 2.97457 | − | 5.15211i | −14.5389 | − | 8.39402i | 45.8003i | 41.9646 | − | 72.6849i | − | 160.481i | 93.8621 | + | 162.574i | 103.419 | |||||
11.6 | −3.47160 | + | 2.00433i | −13.3195 | + | 23.0700i | −7.96531 | + | 13.7963i | 85.3532 | + | 49.2787i | − | 106.786i | −91.8980 | + | 159.172i | − | 192.138i | −233.316 | − | 404.115i | −395.083 | ||||
11.7 | −2.20874 | + | 1.27522i | 14.9045 | − | 25.8153i | −12.7476 | + | 22.0796i | 67.9636 | + | 39.2388i | 76.0257i | 34.5008 | − | 59.7572i | − | 146.638i | −322.785 | − | 559.081i | −200.152 | |||||
11.8 | −1.48416 | + | 0.856882i | −9.17590 | + | 15.8931i | −14.5315 | + | 25.1693i | −29.0287 | − | 16.7598i | − | 31.4507i | 80.8273 | − | 139.997i | − | 104.648i | −46.8942 | − | 81.2231i | 57.4445 | ||||
11.9 | 0.312322 | − | 0.180319i | 9.26592 | − | 16.0490i | −15.9350 | + | 27.6002i | −90.8164 | − | 52.4329i | − | 6.68329i | −48.1376 | + | 83.3769i | 23.0340i | −50.2145 | − | 86.9741i | −37.8186 | |||||
11.10 | 0.987887 | − | 0.570357i | 2.15443 | − | 3.73158i | −15.3494 | + | 26.5859i | 19.1329 | + | 11.0464i | − | 4.91517i | −51.7707 | + | 89.6695i | 71.5213i | 112.217 | + | 194.365i | 25.2015 | |||||
11.11 | 4.33309 | − | 2.50171i | −3.26958 | + | 5.66309i | −3.48290 | + | 6.03256i | 54.4047 | + | 31.4106i | 32.7182i | 22.9091 | − | 39.6798i | 194.962i | 100.120 | + | 173.412i | 314.320 | ||||||
11.12 | 5.18743 | − | 2.99497i | 9.51908 | − | 16.4875i | 1.93964 | − | 3.35955i | −4.99548 | − | 2.88414i | − | 114.037i | 118.181 | − | 204.696i | 168.441i | −59.7259 | − | 103.448i | −34.5516 | |||||
11.13 | 5.44256 | − | 3.14226i | −10.4803 | + | 18.1524i | 3.74762 | − | 6.49106i | −46.0737 | − | 26.6006i | 131.727i | −56.9410 | + | 98.6246i | 154.001i | −98.1728 | − | 170.040i | −334.345 | ||||||
11.14 | 7.81751 | − | 4.51344i | 10.9075 | − | 18.8923i | 24.7424 | − | 42.8550i | 44.4487 | + | 25.6624i | − | 196.921i | −121.740 | + | 210.859i | − | 157.833i | −116.446 | − | 201.690i | 463.304 | ||||
11.15 | 8.27772 | − | 4.77914i | 2.34354 | − | 4.05913i | 29.6804 | − | 51.4080i | −63.0654 | − | 36.4108i | − | 44.8004i | 27.0685 | − | 46.8840i | − | 261.522i | 110.516 | + | 191.419i | −696.050 | ||||
11.16 | 9.35252 | − | 5.39968i | −9.83385 | + | 17.0327i | 42.3131 | − | 73.2884i | 68.3185 | + | 39.4437i | 212.399i | 50.6016 | − | 87.6446i | − | 568.329i | −71.9093 | − | 124.551i | 851.933 | |||||
27.1 | −9.33962 | − | 5.39223i | 14.0993 | + | 24.4208i | 42.1523 | + | 73.0099i | −41.0060 | + | 23.6748i | − | 304.108i | −18.9093 | − | 32.7518i | − | 564.077i | −276.083 | + | 478.189i | 510.641 | ||||
27.2 | −9.18598 | − | 5.30353i | −6.26292 | − | 10.8477i | 40.2548 | + | 69.7233i | 63.9646 | − | 36.9300i | 132.862i | 113.255 | + | 196.163i | − | 514.543i | 43.0517 | − | 74.5678i | −783.436 | |||||
27.3 | −7.60377 | − | 4.39004i | −8.72617 | − | 15.1142i | 22.5449 | + | 39.0489i | −72.9853 | + | 42.1381i | 153.233i | −77.9335 | − | 134.985i | − | 114.930i | −30.7921 | + | 53.3336i | 739.952 | |||||
27.4 | −6.08219 | − | 3.51156i | 3.15654 | + | 5.46728i | 8.66206 | + | 15.0031i | 30.9238 | − | 17.8538i | − | 44.3374i | −47.9780 | − | 83.1003i | 103.070i | 101.573 | − | 175.929i | −250.779 | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.e | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.6.e.a | ✓ | 32 |
37.e | even | 6 | 1 | inner | 37.6.e.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.6.e.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
37.6.e.a | ✓ | 32 | 37.e | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).