Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [109,6,Mod(12,109)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109, base_ring=CyclotomicField(54))
chi = DirichletCharacter(H, H._module([29]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("109.12");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 109 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 109.k (of order \(54\), degree \(18\), 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: | \(17.4818363596\) |
Analytic rank: | \(0\) |
Dimension: | \(810\) |
Relative dimension: | \(45\) over \(\Q(\zeta_{54})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{54}]$ |
$q$-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 | −3.65593 | + | 10.0446i | 26.4787 | − | 6.27556i | −63.0143 | − | 52.8753i | −5.72252 | + | 3.76376i | −33.7688 | + | 288.910i | −0.671442 | − | 11.5282i | 465.258 | − | 268.617i | 444.585 | − | 223.279i | −16.8843 | − | 71.2404i |
12.2 | −3.62689 | + | 9.96480i | −5.02656 | + | 1.19132i | −61.6295 | − | 51.7133i | −49.7432 | + | 32.7166i | 6.35956 | − | 54.4095i | −2.37763 | − | 40.8224i | 444.960 | − | 256.898i | −193.306 | + | 97.0817i | −145.601 | − | 614.341i |
12.3 | −3.51872 | + | 9.66760i | 3.96132 | − | 0.938850i | −56.5677 | − | 47.4659i | 40.2791 | − | 26.4920i | −4.86234 | + | 41.6000i | 11.4295 | + | 196.238i | 372.817 | − | 215.246i | −202.342 | + | 101.620i | 114.383 | + | 482.620i |
12.4 | −3.38854 | + | 9.30995i | −29.6099 | + | 7.01769i | −50.6794 | − | 42.5251i | −18.4132 | + | 12.1106i | 35.0003 | − | 299.447i | 6.36974 | + | 109.364i | 293.073 | − | 169.206i | 610.348 | − | 306.528i | −50.3547 | − | 212.463i |
12.5 | −3.38086 | + | 9.28883i | −14.4411 | + | 3.42259i | −50.3388 | − | 42.2393i | 52.2807 | − | 34.3855i | 17.0313 | − | 145.712i | −11.6256 | − | 199.603i | 288.601 | − | 166.624i | −20.3226 | + | 10.2064i | 142.648 | + | 601.879i |
12.6 | −2.94203 | + | 8.08315i | 14.6409 | − | 3.46996i | −32.1684 | − | 26.9925i | 16.9489 | − | 11.1474i | −15.0258 | + | 128.554i | −7.07504 | − | 121.474i | 74.4419 | − | 42.9791i | −14.8367 | + | 7.45126i | 40.2425 | + | 169.796i |
12.7 | −2.77911 | + | 7.63554i | 13.2124 | − | 3.13139i | −26.0646 | − | 21.8708i | −82.1562 | + | 54.0350i | −12.8088 | + | 109.586i | −11.0768 | − | 190.181i | 14.2496 | − | 8.22703i | −52.3916 | + | 26.3121i | −184.265 | − | 777.476i |
12.8 | −2.76985 | + | 7.61010i | −6.96070 | + | 1.64972i | −25.7281 | − | 21.5885i | −57.0888 | + | 37.5479i | 6.72559 | − | 57.5411i | 8.48020 | + | 145.599i | 11.1212 | − | 6.42083i | −171.423 | + | 86.0918i | −127.616 | − | 538.454i |
12.9 | −2.75736 | + | 7.57578i | 11.2738 | − | 2.67195i | −25.2759 | − | 21.2090i | 68.8677 | − | 45.2950i | −10.8439 | + | 92.7755i | −0.319872 | − | 5.49199i | 6.94979 | − | 4.01246i | −97.1929 | + | 48.8121i | 153.252 | + | 646.621i |
12.10 | −2.40029 | + | 6.59473i | −19.6016 | + | 4.64566i | −13.2157 | − | 11.0893i | 6.30294 | − | 4.14551i | 16.4125 | − | 140.418i | −4.37114 | − | 75.0495i | −89.6352 | + | 51.7509i | 145.487 | − | 73.0662i | 12.2097 | + | 51.5166i |
12.11 | −2.29356 | + | 6.30151i | −19.1346 | + | 4.53497i | −9.93518 | − | 8.33661i | 81.3756 | − | 53.5216i | 15.3091 | − | 130.978i | 10.1737 | + | 174.676i | −110.520 | + | 63.8087i | 128.413 | − | 64.4913i | 150.627 | + | 635.544i |
12.12 | −2.26029 | + | 6.21011i | 21.4109 | − | 5.07447i | −8.94307 | − | 7.50412i | −38.7861 | + | 25.5100i | −16.8819 | + | 144.433i | 9.77571 | + | 167.842i | −116.329 | + | 67.1626i | 215.522 | − | 108.239i | −70.7519 | − | 298.526i |
12.13 | −2.15628 | + | 5.92433i | −10.7516 | + | 2.54819i | −5.93468 | − | 4.97979i | −16.2841 | + | 10.7102i | 8.08725 | − | 69.1909i | 1.48030 | + | 25.4159i | −132.418 | + | 76.4514i | −108.048 | + | 54.2638i | −28.3379 | − | 119.567i |
12.14 | −1.64009 | + | 4.50610i | 28.9642 | − | 6.86463i | 6.89838 | + | 5.78843i | 79.8742 | − | 52.5341i | −16.5710 | + | 141.774i | 0.0919795 | + | 1.57923i | −170.288 | + | 98.3159i | 574.647 | − | 288.599i | 105.723 | + | 446.082i |
12.15 | −1.63205 | + | 4.48401i | −25.4903 | + | 6.04131i | 7.07062 | + | 5.93296i | −84.8211 | + | 55.7877i | 14.5121 | − | 124.159i | −13.6673 | − | 234.658i | −170.383 | + | 98.3705i | 396.106 | − | 198.932i | −111.721 | − | 471.387i |
12.16 | −1.56667 | + | 4.30438i | 4.68011 | − | 1.10921i | 8.44018 | + | 7.08215i | 34.1751 | − | 22.4773i | −2.55772 | + | 21.8827i | −4.41415 | − | 75.7881i | −170.649 | + | 98.5244i | −196.480 | + | 98.6758i | 43.2099 | + | 182.317i |
12.17 | −1.15065 | + | 3.16140i | 7.70306 | − | 1.82566i | 15.8430 | + | 13.2939i | −12.9234 | + | 8.49987i | −3.09193 | + | 26.4531i | −5.68504 | − | 97.6083i | −153.491 | + | 88.6180i | −161.149 | + | 80.9319i | −12.0011 | − | 50.6365i |
12.18 | −0.966472 | + | 2.65536i | −18.2988 | + | 4.33691i | 18.3966 | + | 15.4365i | 33.8630 | − | 22.2720i | 6.16927 | − | 52.7815i | −3.68670 | − | 63.2983i | −137.080 | + | 79.1429i | 98.8862 | − | 49.6625i | 26.4126 | + | 111.444i |
12.19 | −0.718640 | + | 1.97445i | 25.6230 | − | 6.07276i | 21.1314 | + | 17.7314i | −18.6909 | + | 12.2932i | −6.42335 | + | 54.9553i | −12.4742 | − | 214.173i | −108.425 | + | 62.5990i | 402.506 | − | 202.146i | −10.8402 | − | 45.7386i |
12.20 | −0.634883 | + | 1.74433i | 6.41049 | − | 1.51931i | 21.8738 | + | 18.3543i | 44.4897 | − | 29.2614i | −1.41973 | + | 12.1466i | 12.1549 | + | 208.692i | −97.3458 | + | 56.2026i | −178.367 | + | 89.5791i | 22.7956 | + | 96.1821i |
See next 80 embeddings (of 810 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
109.k | even | 54 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 109.6.k.a | ✓ | 810 |
109.k | even | 54 | 1 | inner | 109.6.k.a | ✓ | 810 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
109.6.k.a | ✓ | 810 | 1.a | even | 1 | 1 | trivial |
109.6.k.a | ✓ | 810 | 109.k | even | 54 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(109, [\chi])\).