Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [54,7,Mod(5,54)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(54, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([5]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("54.5");
S:= CuspForms(chi, 7);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 54 = 2 \cdot 3^{3} \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 54.f (of order \(18\), degree \(6\), 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: | \(12.4229205155\) |
Analytic rank: | \(0\) |
Dimension: | \(108\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{18})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −3.63616 | − | 4.33340i | −26.9889 | − | 0.772588i | −5.55674 | + | 31.5138i | 23.6620 | + | 65.0107i | 94.7881 | + | 119.763i | 15.7287 | + | 89.2017i | 156.767 | − | 90.5097i | 727.806 | + | 41.7027i | 195.679 | − | 338.926i |
5.2 | −3.63616 | − | 4.33340i | −18.7528 | − | 19.4250i | −5.55674 | + | 31.5138i | −70.4704 | − | 193.616i | −15.9881 | + | 151.896i | −93.1758 | − | 528.426i | 156.767 | − | 90.5097i | −25.6615 | + | 728.548i | −582.774 | + | 1009.39i |
5.3 | −3.63616 | − | 4.33340i | −15.9087 | + | 21.8154i | −5.55674 | + | 31.5138i | −37.7226 | − | 103.642i | 152.382 | − | 10.3855i | −61.0019 | − | 345.959i | 156.767 | − | 90.5097i | −222.826 | − | 694.111i | −311.957 | + | 540.326i |
5.4 | −3.63616 | − | 4.33340i | −5.10601 | − | 26.5128i | −5.55674 | + | 31.5138i | −10.7717 | − | 29.5950i | −96.3244 | + | 118.531i | 95.4997 | + | 541.606i | 156.767 | − | 90.5097i | −676.857 | + | 270.749i | −89.0795 | + | 154.290i |
5.5 | −3.63616 | − | 4.33340i | −0.750158 | + | 26.9896i | −5.55674 | + | 31.5138i | 81.1361 | + | 222.920i | 119.684 | − | 94.8876i | 8.32537 | + | 47.2155i | 156.767 | − | 90.5097i | −727.875 | − | 40.4929i | 670.977 | − | 1162.17i |
5.6 | −3.63616 | − | 4.33340i | 8.16974 | + | 25.7343i | −5.55674 | + | 31.5138i | −59.7321 | − | 164.113i | 81.8107 | − | 128.977i | 83.7342 | + | 474.880i | 156.767 | − | 90.5097i | −595.511 | + | 420.485i | −493.971 | + | 855.583i |
5.7 | −3.63616 | − | 4.33340i | 17.9291 | − | 20.1878i | −5.55674 | + | 31.5138i | 20.1217 | + | 55.2839i | −152.675 | − | 4.28805i | −42.4874 | − | 240.958i | 156.767 | − | 90.5097i | −86.0942 | − | 723.898i | 166.402 | − | 288.216i |
5.8 | −3.63616 | − | 4.33340i | 23.0877 | + | 13.9985i | −5.55674 | + | 31.5138i | −1.37580 | − | 3.77997i | −23.2893 | − | 150.949i | −65.8860 | − | 373.658i | 156.767 | − | 90.5097i | 337.084 | + | 646.387i | −11.3775 | + | 19.7065i |
5.9 | −3.63616 | − | 4.33340i | 26.8457 | + | 2.88196i | −5.55674 | + | 31.5138i | 6.65021 | + | 18.2713i | −85.1266 | − | 126.813i | 42.6327 | + | 241.782i | 156.767 | − | 90.5097i | 712.389 | + | 154.737i | 54.9957 | − | 95.2554i |
5.10 | 3.63616 | + | 4.33340i | −26.5547 | − | 4.88367i | −5.55674 | + | 31.5138i | −55.9790 | − | 153.801i | −75.3939 | − | 132.830i | 71.8292 | + | 407.363i | −156.767 | + | 90.5097i | 681.299 | + | 259.369i | 462.933 | − | 801.824i |
5.11 | 3.63616 | + | 4.33340i | −18.7421 | + | 19.4354i | −5.55674 | + | 31.5138i | 42.0848 | + | 115.627i | −152.370 | − | 10.5469i | 49.3468 | + | 279.859i | −156.767 | + | 90.5097i | −26.4686 | − | 728.519i | −348.032 | + | 602.809i |
5.12 | 3.63616 | + | 4.33340i | −16.4084 | − | 21.4421i | −5.55674 | + | 31.5138i | 32.0681 | + | 88.1063i | 33.2537 | − | 149.071i | −50.3203 | − | 285.381i | −156.767 | + | 90.5097i | −190.528 | + | 703.662i | −265.196 | + | 459.332i |
5.13 | 3.63616 | + | 4.33340i | −6.50900 | + | 26.2037i | −5.55674 | + | 31.5138i | −50.4685 | − | 138.661i | −137.219 | + | 67.0746i | −64.1010 | − | 363.535i | −156.767 | + | 90.5097i | −644.266 | − | 341.119i | 417.362 | − | 722.893i |
5.14 | 3.63616 | + | 4.33340i | −4.60243 | − | 26.6048i | −5.55674 | + | 31.5138i | −47.7747 | − | 131.260i | 98.5543 | − | 116.684i | 31.0953 | + | 176.350i | −156.767 | + | 90.5097i | −686.635 | + | 244.894i | 395.086 | − | 684.309i |
5.15 | 3.63616 | + | 4.33340i | 12.1966 | − | 24.0882i | −5.55674 | + | 31.5138i | 54.4203 | + | 149.519i | 148.733 | − | 34.7359i | 29.4094 | + | 166.789i | −156.767 | + | 90.5097i | −431.487 | − | 587.589i | −450.043 | + | 779.498i |
5.16 | 3.63616 | + | 4.33340i | 20.8741 | + | 17.1252i | −5.55674 | + | 31.5138i | −41.8999 | − | 115.119i | 1.69101 | + | 152.726i | 68.2247 | + | 386.921i | −156.767 | + | 90.5097i | 142.455 | + | 714.946i | 346.503 | − | 600.160i |
5.17 | 3.63616 | + | 4.33340i | 21.1953 | + | 16.7260i | −5.55674 | + | 31.5138i | 53.1645 | + | 146.068i | 4.58881 | + | 152.666i | −17.0891 | − | 96.9171i | −156.767 | + | 90.5097i | 169.481 | + | 709.026i | −439.658 | + | 761.510i |
5.18 | 3.63616 | + | 4.33340i | 25.1415 | − | 9.84405i | −5.55674 | + | 31.5138i | −34.1182 | − | 93.7391i | 134.077 | + | 73.1537i | −101.765 | − | 577.136i | −156.767 | + | 90.5097i | 535.189 | − | 494.988i | 282.150 | − | 488.698i |
11.1 | −3.63616 | + | 4.33340i | −26.9889 | + | 0.772588i | −5.55674 | − | 31.5138i | 23.6620 | − | 65.0107i | 94.7881 | − | 119.763i | 15.7287 | − | 89.2017i | 156.767 | + | 90.5097i | 727.806 | − | 41.7027i | 195.679 | + | 338.926i |
11.2 | −3.63616 | + | 4.33340i | −18.7528 | + | 19.4250i | −5.55674 | − | 31.5138i | −70.4704 | + | 193.616i | −15.9881 | − | 151.896i | −93.1758 | + | 528.426i | 156.767 | + | 90.5097i | −25.6615 | − | 728.548i | −582.774 | − | 1009.39i |
See next 80 embeddings (of 108 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.f | odd | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 54.7.f.a | ✓ | 108 |
27.f | odd | 18 | 1 | inner | 54.7.f.a | ✓ | 108 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
54.7.f.a | ✓ | 108 | 1.a | even | 1 | 1 | trivial |
54.7.f.a | ✓ | 108 | 27.f | odd | 18 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(54, [\chi])\).