Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,9,Mod(3,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(42))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.3");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 43 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 43.h (of order \(42\), degree \(12\), 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: | \(17.5172802326\) |
Analytic rank: | \(0\) |
Dimension: | \(336\) |
Relative dimension: | \(28\) over \(\Q(\zeta_{42})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{42}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
3.1 | −12.8635 | + | 26.7113i | −30.3876 | − | 2.27723i | −388.410 | − | 487.050i | 136.958 | − | 444.007i | 451.717 | − | 782.398i | 1583.71 | − | 914.357i | 10606.6 | − | 2420.89i | −5569.50 | − | 839.467i | 10098.2 | + | 9369.79i |
3.2 | −12.1501 | + | 25.2300i | 111.026 | + | 8.32026i | −329.314 | − | 412.947i | −217.435 | + | 704.908i | −1558.90 | + | 2700.10i | 939.408 | − | 542.367i | 7430.79 | − | 1696.03i | 5769.85 | + | 869.665i | −15143.0 | − | 14050.6i |
3.3 | −11.8000 | + | 24.5029i | −134.895 | − | 10.1090i | −301.539 | − | 378.118i | −246.719 | + | 799.843i | 1839.46 | − | 3186.03i | 1956.96 | − | 1129.85i | 6035.48 | − | 1377.56i | 11606.8 | + | 1749.44i | −16687.2 | − | 15483.4i |
3.4 | −11.2950 | + | 23.4544i | 75.5829 | + | 5.66415i | −262.917 | − | 329.688i | 125.810 | − | 407.865i | −986.561 | + | 1708.77i | −2233.56 | + | 1289.54i | 4205.09 | − | 959.784i | −807.029 | − | 121.640i | 8145.21 | + | 7557.65i |
3.5 | −10.6351 | + | 22.0841i | −39.8410 | − | 2.98567i | −214.987 | − | 269.585i | −185.498 | + | 601.371i | 489.650 | − | 848.099i | −2744.41 | + | 1584.49i | 2122.34 | − | 484.410i | −4909.33 | − | 739.962i | −11307.9 | − | 10492.2i |
3.6 | −9.45062 | + | 19.6244i | −134.063 | − | 10.0466i | −136.190 | − | 170.777i | 301.256 | − | 976.647i | 1464.14 | − | 2535.96i | −1417.77 | + | 818.552i | −797.764 | + | 182.084i | 11384.2 | + | 1715.90i | 16319.1 | + | 15141.9i |
3.7 | −7.77760 | + | 16.1503i | 155.893 | + | 11.6826i | −40.7292 | − | 51.0728i | 221.849 | − | 719.217i | −1401.15 | + | 2426.87i | 890.825 | − | 514.318i | −3332.26 | + | 760.567i | 17678.6 | + | 2664.61i | 9890.15 | + | 9176.71i |
3.8 | −7.23745 | + | 15.0287i | −0.134245 | − | 0.0100603i | −13.8685 | − | 17.3905i | −60.7674 | + | 197.003i | 1.12279 | − | 1.94472i | 4135.93 | − | 2387.88i | −3801.45 | + | 867.655i | −6487.70 | − | 977.864i | −2520.90 | − | 2339.06i |
3.9 | −6.93493 | + | 14.4005i | 26.0262 | + | 1.95039i | 0.331155 | + | 0.415255i | 249.037 | − | 807.358i | −208.577 | + | 361.265i | 1041.77 | − | 601.469i | −3997.44 | + | 912.389i | −5814.16 | − | 876.344i | 9899.34 | + | 9185.24i |
3.10 | −5.92801 | + | 12.3096i | −79.6637 | − | 5.96997i | 43.2276 | + | 54.2057i | −75.1797 | + | 243.727i | 545.735 | − | 945.241i | −916.434 | + | 529.103i | −4333.45 | + | 989.083i | −177.047 | − | 26.6855i | −2554.52 | − | 2370.25i |
3.11 | −5.27963 | + | 10.9633i | 85.9978 | + | 6.44465i | 67.2948 | + | 84.3850i | −236.441 | + | 766.523i | −524.691 | + | 908.791i | −59.3437 | + | 34.2621i | −4317.41 | + | 985.420i | 866.371 | + | 130.584i | −7155.27 | − | 6639.12i |
3.12 | −1.75641 | + | 3.64723i | 68.2937 | + | 5.11790i | 149.396 | + | 187.337i | 59.6815 | − | 193.483i | −138.618 | + | 240.093i | −4069.77 | + | 2349.69i | −1956.00 | + | 446.443i | −1849.89 | − | 278.826i | 600.850 | + | 557.507i |
3.13 | −1.25301 | + | 2.60190i | −146.817 | − | 11.0024i | 154.414 | + | 193.628i | −60.6007 | + | 196.463i | 212.590 | − | 368.217i | 3053.32 | − | 1762.84i | −1418.05 | + | 323.660i | 14946.6 | + | 2252.83i | −435.242 | − | 403.846i |
3.14 | −0.385973 | + | 0.801481i | −82.9635 | − | 6.21726i | 159.120 | + | 199.530i | 153.970 | − | 499.159i | 37.0047 | − | 64.0940i | −701.273 | + | 404.880i | −443.358 | + | 101.194i | 356.569 | + | 53.7442i | 340.638 | + | 316.066i |
3.15 | 0.864743 | − | 1.79566i | 1.37697 | + | 0.103190i | 157.137 | + | 197.043i | 286.537 | − | 928.929i | 1.37602 | − | 2.38334i | 941.195 | − | 543.399i | 987.128 | − | 225.306i | −6485.83 | − | 977.582i | −1420.26 | − | 1317.81i |
3.16 | 1.44533 | − | 3.00125i | −54.6425 | − | 4.09489i | 152.695 | + | 191.473i | −330.560 | + | 1071.65i | −91.2660 | + | 158.077i | −764.563 | + | 441.421i | 1626.74 | − | 371.294i | −3518.69 | − | 530.357i | 2738.52 | + | 2540.98i |
3.17 | 1.83542 | − | 3.81129i | 121.748 | + | 9.12376i | 148.456 | + | 186.158i | 21.4535 | − | 69.5504i | 258.232 | − | 447.271i | 1381.02 | − | 797.333i | 2037.77 | − | 465.107i | 8251.65 | + | 1243.74i | −225.701 | − | 209.420i |
3.18 | 3.70483 | − | 7.69316i | 20.6613 | + | 1.54835i | 114.154 | + | 143.145i | −181.686 | + | 589.011i | 88.4583 | − | 153.214i | 2375.79 | − | 1371.67i | 3655.28 | − | 834.293i | −6063.23 | − | 913.884i | 3858.24 | + | 3579.93i |
3.19 | 6.51335 | − | 13.5251i | −137.776 | − | 10.3249i | 19.1085 | + | 23.9613i | −3.15223 | + | 10.2193i | −1037.03 | + | 1796.19i | −2870.44 | + | 1657.25i | 4195.19 | − | 957.525i | 12387.9 | + | 1867.18i | 117.685 | + | 109.196i |
3.20 | 7.08079 | − | 14.7034i | −17.7574 | − | 1.33074i | −6.43935 | − | 8.07469i | 52.6500 | − | 170.687i | −145.303 | + | 251.672i | −2020.79 | + | 1166.70i | 3908.74 | − | 892.144i | −6174.16 | − | 930.605i | −2136.88 | − | 1982.73i |
See next 80 embeddings (of 336 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
43.h | odd | 42 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 43.9.h.a | ✓ | 336 |
43.h | odd | 42 | 1 | inner | 43.9.h.a | ✓ | 336 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
43.9.h.a | ✓ | 336 | 1.a | even | 1 | 1 | trivial |
43.9.h.a | ✓ | 336 | 43.h | odd | 42 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(43, [\chi])\).