Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [572,2,Mod(9,572)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(30))
chi = DirichletCharacter(H, H._module([0, 18, 20]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("572.9");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 572 = 2^{2} \cdot 11 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 572.bg (of order \(15\), degree \(8\), 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: | \(4.56744299562\) |
Analytic rank: | \(0\) |
Dimension: | \(112\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{15})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{15}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | 0 | −0.351299 | − | 3.34239i | 0 | −0.419191 | − | 1.29014i | 0 | −0.149591 | + | 1.42326i | 0 | −8.11369 | + | 1.72462i | 0 | ||||||||||
9.2 | 0 | −0.263130 | − | 2.50351i | 0 | 1.21356 | + | 3.73495i | 0 | 0.197160 | − | 1.87585i | 0 | −3.26389 | + | 0.693762i | 0 | ||||||||||
9.3 | 0 | −0.229534 | − | 2.18387i | 0 | −0.547837 | − | 1.68607i | 0 | −0.0708172 | + | 0.673781i | 0 | −1.78216 | + | 0.378809i | 0 | ||||||||||
9.4 | 0 | −0.165782 | − | 1.57731i | 0 | −1.14854 | − | 3.53484i | 0 | 0.493072 | − | 4.69127i | 0 | 0.474019 | − | 0.100756i | 0 | ||||||||||
9.5 | 0 | −0.153632 | − | 1.46171i | 0 | 0.347814 | + | 1.07046i | 0 | 0.456258 | − | 4.34101i | 0 | 0.821453 | − | 0.174605i | 0 | ||||||||||
9.6 | 0 | −0.0694093 | − | 0.660385i | 0 | −0.596743 | − | 1.83659i | 0 | −0.373949 | + | 3.55789i | 0 | 2.50315 | − | 0.532061i | 0 | ||||||||||
9.7 | 0 | −0.0692906 | − | 0.659256i | 0 | 0.701430 | + | 2.15878i | 0 | −0.337098 | + | 3.20727i | 0 | 2.50463 | − | 0.532375i | 0 | ||||||||||
9.8 | 0 | 0.0211820 | + | 0.201533i | 0 | −0.370496 | − | 1.14027i | 0 | −0.131751 | + | 1.25353i | 0 | 2.89428 | − | 0.615197i | 0 | ||||||||||
9.9 | 0 | 0.0911637 | + | 0.867364i | 0 | 0.671959 | + | 2.06808i | 0 | 0.257245 | − | 2.44753i | 0 | 2.19043 | − | 0.465591i | 0 | ||||||||||
9.10 | 0 | 0.151007 | + | 1.43674i | 0 | −0.310916 | − | 0.956900i | 0 | 0.0904890 | − | 0.860946i | 0 | 0.893027 | − | 0.189819i | 0 | ||||||||||
9.11 | 0 | 0.161126 | + | 1.53301i | 0 | 1.30886 | + | 4.02825i | 0 | −0.295810 | + | 2.81444i | 0 | 0.610280 | − | 0.129719i | 0 | ||||||||||
9.12 | 0 | 0.251214 | + | 2.39014i | 0 | −0.900429 | − | 2.77123i | 0 | 0.251542 | − | 2.39326i | 0 | −2.71524 | + | 0.577141i | 0 | ||||||||||
9.13 | 0 | 0.277675 | + | 2.64191i | 0 | 0.389171 | + | 1.19775i | 0 | 0.0870458 | − | 0.828185i | 0 | −3.96812 | + | 0.843450i | 0 | ||||||||||
9.14 | 0 | 0.348708 | + | 3.31773i | 0 | −0.338638 | − | 1.04222i | 0 | −0.473796 | + | 4.50786i | 0 | −7.95129 | + | 1.69010i | 0 | ||||||||||
81.1 | 0 | −2.59967 | + | 0.552578i | 0 | −1.82163 | + | 1.32349i | 0 | −4.40077 | − | 0.935412i | 0 | 3.71233 | − | 1.65283i | 0 | ||||||||||
81.2 | 0 | −2.59692 | + | 0.551993i | 0 | 1.71674 | − | 1.24728i | 0 | −1.20270 | − | 0.255642i | 0 | 3.69867 | − | 1.64675i | 0 | ||||||||||
81.3 | 0 | −2.50220 | + | 0.531859i | 0 | −0.767747 | + | 0.557801i | 0 | 3.31034 | + | 0.703635i | 0 | 3.23748 | − | 1.44142i | 0 | ||||||||||
81.4 | 0 | −2.03406 | + | 0.432352i | 0 | 3.08678 | − | 2.24268i | 0 | 1.74512 | + | 0.370936i | 0 | 1.20982 | − | 0.538647i | 0 | ||||||||||
81.5 | 0 | −0.978574 | + | 0.208002i | 0 | −3.20054 | + | 2.32533i | 0 | 0.357037 | + | 0.0758905i | 0 | −1.82629 | + | 0.813119i | 0 | ||||||||||
81.6 | 0 | −0.807770 | + | 0.171697i | 0 | 0.437086 | − | 0.317562i | 0 | −2.23753 | − | 0.475601i | 0 | −2.11762 | + | 0.942827i | 0 | ||||||||||
See next 80 embeddings (of 112 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.c | even | 5 | 1 | inner |
13.c | even | 3 | 1 | inner |
143.q | even | 15 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 572.2.bg.a | ✓ | 112 |
11.c | even | 5 | 1 | inner | 572.2.bg.a | ✓ | 112 |
13.c | even | 3 | 1 | inner | 572.2.bg.a | ✓ | 112 |
143.q | even | 15 | 1 | inner | 572.2.bg.a | ✓ | 112 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
572.2.bg.a | ✓ | 112 | 1.a | even | 1 | 1 | trivial |
572.2.bg.a | ✓ | 112 | 11.c | even | 5 | 1 | inner |
572.2.bg.a | ✓ | 112 | 13.c | even | 3 | 1 | inner |
572.2.bg.a | ✓ | 112 | 143.q | even | 15 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).