Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,5,Mod(3,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(32))
chi = DirichletCharacter(H, H._module([16, 3]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.3");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 128.l (of order \(32\), degree \(16\), 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: | \(13.2313552747\) |
Analytic rank: | \(0\) |
Dimension: | \(1008\) |
Relative dimension: | \(63\) over \(\Q(\zeta_{32})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{32}]$ |
$q$-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 | −3.98767 | + | 0.313821i | −13.0973 | + | 10.7487i | 15.8030 | − | 2.50283i | 37.0978 | + | 11.2535i | 48.8546 | − | 46.9724i | 4.64899 | + | 23.3721i | −62.2319 | + | 14.9398i | 40.2030 | − | 202.114i | −151.465 | − | 33.2332i |
3.2 | −3.95882 | − | 0.572521i | −4.36463 | + | 3.58196i | 15.3444 | + | 4.53301i | 2.00607 | + | 0.608536i | 19.3295 | − | 11.6815i | −7.49964 | − | 37.7032i | −58.1506 | − | 26.7304i | −9.58275 | + | 48.1757i | −7.59328 | − | 3.55760i |
3.3 | −3.93898 | + | 0.696008i | 6.46187 | − | 5.30312i | 15.0311 | − | 5.48312i | −28.0844 | − | 8.51932i | −21.7622 | + | 25.3864i | 0.922571 | + | 4.63808i | −55.3911 | + | 32.0597i | −2.16964 | + | 10.9075i | 116.554 | + | 14.0104i |
3.4 | −3.93799 | − | 0.701605i | 11.4156 | − | 9.36851i | 15.0155 | + | 5.52582i | 39.4094 | + | 11.9547i | −51.5273 | + | 28.8839i | −18.6869 | − | 93.9456i | −55.2539 | − | 32.2956i | 26.7438 | − | 134.450i | −146.806 | − | 74.7273i |
3.5 | −3.89182 | + | 0.923973i | −7.47268 | + | 6.13267i | 14.2925 | − | 7.19188i | −34.4748 | − | 10.4578i | 23.4159 | − | 30.7718i | −4.53604 | − | 22.8042i | −48.9789 | + | 41.1954i | 2.42901 | − | 12.2115i | 143.832 | + | 8.84616i |
3.6 | −3.87913 | − | 0.975872i | 13.1338 | − | 10.7787i | 14.0953 | + | 7.57107i | −12.6494 | − | 3.83717i | −61.4665 | + | 28.9949i | 15.2716 | + | 76.7754i | −47.2893 | − | 43.1245i | 40.5159 | − | 203.687i | 45.3243 | + | 27.2291i |
3.7 | −3.87689 | + | 0.984756i | 3.81952 | − | 3.13460i | 14.0605 | − | 7.63558i | 27.8176 | + | 8.43838i | −11.7210 | + | 15.9138i | 7.96282 | + | 40.0318i | −46.9918 | + | 43.4484i | −11.0393 | + | 55.4983i | −116.155 | − | 5.32109i |
3.8 | −3.86190 | − | 1.04198i | 1.98252 | − | 1.62701i | 13.8286 | + | 8.04803i | −2.07933 | − | 0.630759i | −9.35161 | + | 4.21762i | −2.78288 | − | 13.9905i | −45.0187 | − | 45.4898i | −14.5191 | + | 72.9924i | 7.37294 | + | 4.60255i |
3.9 | −3.81656 | − | 1.19746i | −9.21046 | + | 7.55883i | 13.1322 | + | 9.14031i | −23.7963 | − | 7.21854i | 44.2036 | − | 17.8196i | 16.4786 | + | 82.8433i | −39.1746 | − | 50.6098i | 11.8943 | − | 59.7969i | 82.1761 | + | 56.0450i |
3.10 | −3.38069 | − | 2.13798i | −2.22632 | + | 1.82709i | 6.85808 | + | 14.4557i | 21.3910 | + | 6.48890i | 11.4328 | − | 1.41700i | 12.9621 | + | 65.1649i | 7.72094 | − | 63.5326i | −14.1841 | + | 71.3082i | −58.4433 | − | 67.6706i |
3.11 | −3.37653 | + | 2.14454i | −6.28903 | + | 5.16127i | 6.80190 | − | 14.4822i | 17.4865 | + | 5.30446i | 10.1665 | − | 30.9143i | −14.2328 | − | 71.5529i | 8.09084 | + | 63.4865i | −2.88917 | + | 14.5249i | −70.4191 | + | 19.5897i |
3.12 | −3.26315 | + | 2.31341i | 9.84830 | − | 8.08229i | 5.29629 | − | 15.0980i | −14.5438 | − | 4.41182i | −13.4388 | + | 49.1569i | −8.09349 | − | 40.6887i | 17.6452 | + | 61.5195i | 15.8633 | − | 79.7501i | 57.6650 | − | 19.2494i |
3.13 | −3.25888 | − | 2.31942i | 5.81494 | − | 4.77220i | 5.24061 | + | 15.1174i | −28.0812 | − | 8.51835i | −30.0189 | + | 2.06475i | −9.11012 | − | 45.7997i | 17.9850 | − | 61.4210i | −4.76266 | + | 23.9435i | 71.7558 | + | 92.8924i |
3.14 | −3.22267 | + | 2.36947i | −7.20596 | + | 5.91378i | 4.77123 | − | 15.2720i | 2.33003 | + | 0.706808i | 9.20993 | − | 36.1325i | 11.1538 | + | 56.0737i | 20.8105 | + | 60.5221i | 1.15079 | − | 5.78540i | −9.18370 | + | 3.24313i |
3.15 | −3.09156 | − | 2.53816i | −12.3327 | + | 10.1212i | 3.11546 | + | 15.6938i | −15.2168 | − | 4.61597i | 63.8165 | + | 0.0121723i | −11.6720 | − | 58.6789i | 30.2017 | − | 56.4257i | 33.8548 | − | 170.200i | 35.3276 | + | 52.8933i |
3.16 | −2.96812 | − | 2.68147i | −5.39383 | + | 4.42660i | 1.61947 | + | 15.9178i | 42.9519 | + | 13.0293i | 27.8793 | + | 1.32469i | −8.38667 | − | 42.1626i | 37.8764 | − | 51.5886i | −6.30371 | + | 31.6909i | −92.5487 | − | 153.847i |
3.17 | −2.80758 | + | 2.84912i | −0.236785 | + | 0.194325i | −0.235026 | − | 15.9983i | −27.1487 | − | 8.23546i | 0.111138 | − | 1.22021i | 13.1413 | + | 66.0660i | 46.2409 | + | 44.2468i | −15.7840 | + | 79.3516i | 99.6858 | − | 54.2283i |
3.18 | −2.66833 | − | 2.97993i | 8.56994 | − | 7.03317i | −1.75999 | + | 15.9029i | 16.1113 | + | 4.88731i | −43.8258 | − | 6.77100i | 8.61826 | + | 43.3269i | 52.0858 | − | 37.1896i | 8.17609 | − | 41.1040i | −28.4265 | − | 61.0516i |
3.19 | −2.35233 | + | 3.23520i | 1.64666 | − | 1.35138i | −4.93309 | − | 15.2205i | 31.2429 | + | 9.47742i | 0.498498 | + | 8.50617i | −6.24158 | − | 31.3785i | 60.8458 | + | 19.8442i | −14.9171 | + | 74.9931i | −104.155 | + | 78.7830i |
3.20 | −2.15707 | − | 3.36854i | −1.54548 | + | 1.26834i | −6.69413 | + | 14.5323i | −43.0952 | − | 13.0728i | 7.60615 | + | 2.47011i | 10.6247 | + | 53.4142i | 63.3924 | − | 8.79776i | −15.0225 | + | 75.5232i | 48.9230 | + | 173.367i |
See next 80 embeddings (of 1008 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
128.l | odd | 32 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 128.5.l.a | ✓ | 1008 |
128.l | odd | 32 | 1 | inner | 128.5.l.a | ✓ | 1008 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
128.5.l.a | ✓ | 1008 | 1.a | even | 1 | 1 | trivial |
128.5.l.a | ✓ | 1008 | 128.l | odd | 32 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(128, [\chi])\).