Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [37,10,Mod(10,37)]
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([4]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("37.10");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 37 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 37.c (of order \(3\), 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: | \(19.0563259381\) |
Analytic rank: | \(0\) |
Dimension: | \(54\) |
Relative dimension: | \(27\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10.1 | −21.2153 | + | 36.7459i | 86.0505 | + | 149.044i | −644.175 | − | 1115.74i | 1139.39 | + | 1973.48i | −7302.33 | 4123.02 | + | 7141.28i | 32941.0 | −4967.86 | + | 8604.59i | −96689.7 | ||||||
10.2 | −20.2191 | + | 35.0205i | −84.5687 | − | 146.477i | −561.626 | − | 972.764i | −198.799 | − | 344.329i | 6839.62 | 3839.21 | + | 6649.70i | 24717.9 | −4462.23 | + | 7728.80i | 16078.1 | ||||||
10.3 | −19.4882 | + | 33.7545i | −36.9005 | − | 63.9135i | −503.578 | − | 872.222i | −226.455 | − | 392.232i | 2876.49 | −4356.37 | − | 7545.46i | 19299.4 | 7118.21 | − | 12329.1i | 17652.8 | ||||||
10.4 | −18.3336 | + | 31.7547i | 82.9316 | + | 143.642i | −416.240 | − | 720.949i | −705.543 | − | 1222.04i | −6081.73 | −777.430 | − | 1346.55i | 11751.1 | −3913.79 | + | 6778.88i | 51740.5 | ||||||
10.5 | −16.2809 | + | 28.1993i | −30.6895 | − | 53.1557i | −274.135 | − | 474.815i | 1243.37 | + | 2153.58i | 1998.61 | −3568.14 | − | 6180.20i | 1181.00 | 7957.81 | − | 13783.3i | −80972.6 | ||||||
10.6 | −11.9270 | + | 20.6581i | 13.2219 | + | 22.9010i | −28.5049 | − | 49.3720i | −269.425 | − | 466.658i | −630.788 | 2392.52 | + | 4143.97i | −10853.3 | 9491.86 | − | 16440.4i | 12853.7 | ||||||
10.7 | −11.2222 | + | 19.4374i | −124.325 | − | 215.338i | 4.12627 | + | 7.14691i | 372.627 | + | 645.409i | 5580.79 | −110.128 | − | 190.748i | −11676.7 | −21072.0 | + | 36497.9i | −16726.7 | ||||||
10.8 | −10.1826 | + | 17.6368i | 107.307 | + | 185.861i | 48.6277 | + | 84.2257i | 603.246 | + | 1044.85i | −4370.67 | −3014.80 | − | 5221.78i | −12407.7 | −13188.1 | + | 22842.4i | −24570.5 | ||||||
10.9 | −9.89842 | + | 17.1446i | 16.8051 | + | 29.1073i | 60.0426 | + | 103.997i | 507.953 | + | 879.801i | −665.375 | 4705.54 | + | 8150.24i | −12513.3 | 9276.68 | − | 16067.7i | −20111.7 | ||||||
10.10 | −9.23482 | + | 15.9952i | −68.0812 | − | 117.920i | 85.4363 | + | 147.980i | −1305.65 | − | 2261.46i | 2514.87 | −1262.57 | − | 2186.84i | −12612.4 | 571.399 | − | 989.692i | 48229.8 | ||||||
10.11 | −3.26260 | + | 5.65099i | 122.847 | + | 212.777i | 234.711 | + | 406.531i | −940.627 | − | 1629.21i | −1603.20 | 3055.33 | + | 5291.98i | −6403.98 | −20341.1 | + | 35231.9i | 12275.6 | ||||||
10.12 | −1.64814 | + | 2.85467i | 22.5023 | + | 38.9752i | 250.567 | + | 433.995i | −142.433 | − | 246.702i | −148.348 | −5192.34 | − | 8993.40i | −3339.58 | 8828.79 | − | 15291.9i | 939.004 | ||||||
10.13 | −0.349842 | + | 0.605944i | −41.9286 | − | 72.6225i | 255.755 | + | 442.981i | 954.706 | + | 1653.60i | 58.6736 | 1656.50 | + | 2869.15i | −716.134 | 6325.48 | − | 10956.1i | −1335.99 | ||||||
10.14 | 0.152284 | − | 0.263764i | −87.8341 | − | 152.133i | 255.954 | + | 443.325i | 209.008 | + | 362.012i | −53.5031 | −2618.01 | − | 4534.52i | 311.850 | −5588.16 | + | 9678.98i | 127.314 | ||||||
10.15 | 3.41387 | − | 5.91300i | 41.2346 | + | 71.4204i | 232.691 | + | 403.033i | −741.590 | − | 1284.47i | 563.079 | 721.322 | + | 1249.37i | 6673.32 | 6440.92 | − | 11156.0i | −10126.8 | ||||||
10.16 | 3.95661 | − | 6.85305i | 96.3161 | + | 166.824i | 224.690 | + | 389.175i | 891.239 | + | 1543.67i | 1524.34 | 2975.14 | + | 5153.10i | 7607.62 | −8712.07 | + | 15089.8i | 14105.1 | ||||||
10.17 | 4.81916 | − | 8.34702i | −93.7353 | − | 162.354i | 209.551 | + | 362.954i | −535.707 | − | 927.872i | −1806.90 | 5666.05 | + | 9813.89i | 8974.26 | −7731.11 | + | 13390.7i | −10326.6 | ||||||
10.18 | 10.3662 | − | 17.9548i | 86.3103 | + | 149.494i | 41.0839 | + | 71.1595i | 563.169 | + | 975.437i | 3578.84 | −1948.40 | − | 3374.73i | 12318.5 | −5057.43 | + | 8759.72i | 23351.7 | ||||||
10.19 | 11.0930 | − | 19.2137i | 11.3281 | + | 19.6208i | 9.88860 | + | 17.1276i | −837.494 | − | 1450.58i | 502.652 | 1133.95 | + | 1964.06i | 11798.1 | 9584.85 | − | 16601.4i | −37161.4 | ||||||
10.20 | 11.7918 | − | 20.4241i | −110.187 | − | 190.850i | −22.0953 | − | 38.2702i | −512.092 | − | 886.969i | −5197.24 | −4542.03 | − | 7867.02i | 11032.7 | −14440.9 | + | 25012.4i | −24154.0 | ||||||
See all 54 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 37.10.c.a | ✓ | 54 |
37.c | even | 3 | 1 | inner | 37.10.c.a | ✓ | 54 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
37.10.c.a | ✓ | 54 | 1.a | even | 1 | 1 | trivial |
37.10.c.a | ✓ | 54 | 37.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).