Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [89,9,Mod(12,89)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(89, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("89.12");
S:= CuspForms(chi, 9);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 89 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 89.d (of order \(8\), degree \(4\), 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: | \(36.2566962954\) |
Analytic rank: | \(0\) |
Dimension: | \(236\) |
Relative dimension: | \(59\) over \(\Q(\zeta_{8})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12.1 | −31.0946 | −9.36236 | − | 22.6027i | 710.877 | −857.445 | − | 857.445i | 291.119 | + | 702.824i | 628.442 | + | 1517.19i | −14144.2 | 4216.10 | − | 4216.10i | 26662.0 | + | 26662.0i | ||||||
12.2 | −31.0011 | 31.3265 | + | 75.6288i | 705.071 | 51.8392 | + | 51.8392i | −971.157 | − | 2344.58i | −917.976 | − | 2216.19i | −13921.7 | −99.0429 | + | 99.0429i | −1607.07 | − | 1607.07i | ||||||
12.3 | −28.9464 | −19.8602 | − | 47.9467i | 581.896 | 735.914 | + | 735.914i | 574.881 | + | 1387.89i | 252.151 | + | 608.747i | −9433.52 | 2734.87 | − | 2734.87i | −21302.1 | − | 21302.1i | ||||||
12.4 | −28.5712 | −56.6127 | − | 136.675i | 560.314 | −23.2205 | − | 23.2205i | 1617.49 | + | 3904.98i | 166.667 | + | 402.371i | −8694.63 | −10835.8 | + | 10835.8i | 663.439 | + | 663.439i | ||||||
12.5 | −27.3135 | −25.5602 | − | 61.7079i | 490.027 | −165.674 | − | 165.674i | 698.140 | + | 1685.46i | −1577.37 | − | 3808.10i | −6392.10 | 1484.79 | − | 1484.79i | 4525.14 | + | 4525.14i | ||||||
12.6 | −26.9582 | 8.42763 | + | 20.3461i | 470.746 | 145.579 | + | 145.579i | −227.194 | − | 548.495i | 1430.71 | + | 3454.05i | −5789.16 | 4296.39 | − | 4296.39i | −3924.54 | − | 3924.54i | ||||||
12.7 | −26.3802 | 46.7395 | + | 112.839i | 439.914 | 143.497 | + | 143.497i | −1233.00 | − | 2976.71i | −358.362 | − | 865.163i | −4851.68 | −5908.74 | + | 5908.74i | −3785.47 | − | 3785.47i | ||||||
12.8 | −26.2674 | 26.2029 | + | 63.2594i | 433.974 | 150.875 | + | 150.875i | −688.281 | − | 1661.66i | 1068.50 | + | 2579.60i | −4674.91 | 1324.17 | − | 1324.17i | −3963.09 | − | 3963.09i | ||||||
12.9 | −25.6656 | 58.9224 | + | 142.251i | 402.721 | −721.025 | − | 721.025i | −1512.28 | − | 3650.96i | 588.391 | + | 1420.50i | −3765.67 | −12124.2 | + | 12124.2i | 18505.5 | + | 18505.5i | ||||||
12.10 | −23.4949 | −24.5114 | − | 59.1758i | 296.008 | −339.543 | − | 339.543i | 575.892 | + | 1390.33i | 235.030 | + | 567.412i | −939.987 | 1738.36 | − | 1738.36i | 7977.52 | + | 7977.52i | ||||||
12.11 | −20.9644 | 33.0521 | + | 79.7947i | 183.506 | 775.587 | + | 775.587i | −692.916 | − | 1672.85i | −1234.34 | − | 2979.97i | 1519.80 | −635.433 | + | 635.433i | −16259.7 | − | 16259.7i | ||||||
12.12 | −20.7331 | −27.2807 | − | 65.8615i | 173.863 | 410.068 | + | 410.068i | 565.615 | + | 1365.52i | −959.586 | − | 2316.64i | 1702.97 | 1045.83 | − | 1045.83i | −8501.98 | − | 8501.98i | ||||||
12.13 | −20.5596 | 15.9072 | + | 38.4034i | 166.695 | −384.710 | − | 384.710i | −327.045 | − | 789.556i | −987.510 | − | 2384.06i | 1836.06 | 3417.55 | − | 3417.55i | 7909.46 | + | 7909.46i | ||||||
12.14 | −18.9016 | 18.5519 | + | 44.7883i | 101.272 | −670.857 | − | 670.857i | −350.662 | − | 846.573i | −29.3264 | − | 70.8003i | 2924.61 | 2977.51 | − | 2977.51i | 12680.3 | + | 12680.3i | ||||||
12.15 | −18.6788 | −38.3619 | − | 92.6139i | 92.8993 | −257.482 | − | 257.482i | 716.557 | + | 1729.92i | 1454.22 | + | 3510.79i | 3046.53 | −2466.37 | + | 2466.37i | 4809.47 | + | 4809.47i | ||||||
12.16 | −18.3903 | 55.4418 | + | 133.848i | 82.2026 | 596.658 | + | 596.658i | −1019.59 | − | 2461.51i | 1012.39 | + | 2444.13i | 3196.18 | −10202.3 | + | 10202.3i | −10972.7 | − | 10972.7i | ||||||
12.17 | −16.0085 | −51.6553 | − | 124.707i | 0.272565 | 637.942 | + | 637.942i | 826.924 | + | 1996.37i | 501.601 | + | 1210.97i | 4093.82 | −8244.21 | + | 8244.21i | −10212.5 | − | 10212.5i | ||||||
12.18 | −13.9550 | −1.45574 | − | 3.51448i | −61.2570 | 488.794 | + | 488.794i | 20.3150 | + | 49.0447i | 318.064 | + | 767.876i | 4427.33 | 4629.10 | − | 4629.10i | −6821.14 | − | 6821.14i | ||||||
12.19 | −13.8988 | −56.4369 | − | 136.251i | −62.8227 | −785.745 | − | 785.745i | 784.406 | + | 1893.72i | −1084.96 | − | 2619.32i | 4431.26 | −10739.8 | + | 10739.8i | 10920.9 | + | 10920.9i | ||||||
12.20 | −12.8757 | 8.12224 | + | 19.6088i | −90.2159 | 332.420 | + | 332.420i | −104.580 | − | 252.478i | 865.626 | + | 2089.81i | 4457.78 | 4320.79 | − | 4320.79i | −4280.15 | − | 4280.15i | ||||||
See next 80 embeddings (of 236 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
89.d | odd | 8 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 89.9.d.a | ✓ | 236 |
89.d | odd | 8 | 1 | inner | 89.9.d.a | ✓ | 236 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
89.9.d.a | ✓ | 236 | 1.a | even | 1 | 1 | trivial |
89.9.d.a | ✓ | 236 | 89.d | odd | 8 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(89, [\chi])\).