Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,3,Mod(28,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 7]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.28");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.r (of order \(20\), 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: | \(6.13080594811\) |
Analytic rank: | \(0\) |
Dimension: | \(80\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{20})\) |
Twist minimal: | no (minimal twist has level 75) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{20}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
28.1 | −1.73350 | − | 3.40219i | 0 | −6.21871 | + | 8.55932i | −2.10687 | + | 4.53443i | 0 | 7.30480 | − | 7.30480i | 24.8151 | + | 3.93033i | 0 | 19.0793 | − | 0.692468i | ||||||
28.2 | −1.51991 | − | 2.98300i | 0 | −4.23700 | + | 5.83172i | −3.83005 | − | 3.21415i | 0 | −3.58606 | + | 3.58606i | 10.6092 | + | 1.68033i | 0 | −3.76645 | + | 16.3102i | ||||||
28.3 | −1.03753 | − | 2.03626i | 0 | −0.718759 | + | 0.989286i | 3.06942 | − | 3.94698i | 0 | 0.410561 | − | 0.410561i | −6.26868 | − | 0.992861i | 0 | −11.2217 | − | 2.15504i | ||||||
28.4 | −0.670302 | − | 1.31554i | 0 | 1.06979 | − | 1.47245i | −0.953987 | + | 4.90815i | 0 | 0.393993 | − | 0.393993i | −8.48731 | − | 1.34426i | 0 | 7.09633 | − | 2.03493i | ||||||
28.5 | −0.421583 | − | 0.827404i | 0 | 1.84428 | − | 2.53843i | −4.83080 | + | 1.28971i | 0 | −4.18270 | + | 4.18270i | −6.54656 | − | 1.03687i | 0 | 3.10370 | + | 3.45331i | ||||||
28.6 | −0.359880 | − | 0.706305i | 0 | 1.98179 | − | 2.72770i | 0.990973 | − | 4.90081i | 0 | 8.25978 | − | 8.25978i | −5.77157 | − | 0.914128i | 0 | −3.81810 | + | 1.06378i | ||||||
28.7 | 0.775767 | + | 1.52253i | 0 | 0.634862 | − | 0.873813i | −4.50160 | + | 2.17615i | 0 | −3.94736 | + | 3.94736i | 8.57385 | + | 1.35797i | 0 | −6.80544 | − | 5.16562i | ||||||
28.8 | 1.07322 | + | 2.10631i | 0 | −0.933591 | + | 1.28498i | −4.10427 | − | 2.85569i | 0 | 9.10511 | − | 9.10511i | 5.63093 | + | 0.891852i | 0 | 1.61020 | − | 11.7096i | ||||||
28.9 | 1.13229 | + | 2.22224i | 0 | −1.30514 | + | 1.79637i | 4.95217 | + | 0.689916i | 0 | 3.10232 | − | 3.10232i | 4.38372 | + | 0.694313i | 0 | 4.07413 | + | 11.7861i | ||||||
28.10 | 1.47736 | + | 2.89947i | 0 | −3.87323 | + | 5.33104i | 3.93419 | − | 3.08579i | 0 | −8.41509 | + | 8.41509i | −8.32297 | − | 1.31823i | 0 | 14.7594 | + | 6.84828i | ||||||
37.1 | −0.607139 | − | 3.83332i | 0 | −10.5215 | + | 3.41865i | −2.58662 | − | 4.27895i | 0 | −2.30555 | − | 2.30555i | 12.4449 | + | 24.4245i | 0 | −14.8321 | + | 12.5133i | ||||||
37.2 | −0.495586 | − | 3.12901i | 0 | −5.74087 | + | 1.86532i | 4.93946 | − | 0.775740i | 0 | 1.18776 | + | 1.18776i | 2.92871 | + | 5.74792i | 0 | −4.87523 | − | 15.0712i | ||||||
37.3 | −0.373594 | − | 2.35878i | 0 | −1.62005 | + | 0.526386i | −4.74635 | − | 1.57230i | 0 | −5.77573 | − | 5.77573i | −2.48998 | − | 4.88686i | 0 | −1.93550 | + | 11.7830i | ||||||
37.4 | −0.229020 | − | 1.44598i | 0 | 1.76583 | − | 0.573753i | 2.10186 | − | 4.53676i | 0 | 8.32198 | + | 8.32198i | −3.89261 | − | 7.63968i | 0 | −7.04141 | − | 2.00024i | ||||||
37.5 | −0.167706 | − | 1.05885i | 0 | 2.71118 | − | 0.880917i | −4.35927 | + | 2.44883i | 0 | −2.58808 | − | 2.58808i | −3.33424 | − | 6.54382i | 0 | 3.32403 | + | 4.20513i | ||||||
37.6 | −0.0337531 | − | 0.213109i | 0 | 3.75995 | − | 1.22168i | 2.09142 | + | 4.54158i | 0 | 2.53517 | + | 2.53517i | −0.779083 | − | 1.52904i | 0 | 0.897260 | − | 0.598992i | ||||||
37.7 | 0.202100 | + | 1.27601i | 0 | 2.21687 | − | 0.720305i | −1.23939 | − | 4.84396i | 0 | 1.27981 | + | 1.27981i | 3.71321 | + | 7.28759i | 0 | 5.93045 | − | 2.56044i | ||||||
37.8 | 0.224472 | + | 1.41726i | 0 | 1.84598 | − | 0.599796i | 3.76307 | + | 3.29231i | 0 | −1.57196 | − | 1.57196i | 3.87022 | + | 7.59573i | 0 | −3.82136 | + | 6.07229i | ||||||
37.9 | 0.479602 | + | 3.02809i | 0 | −5.13507 | + | 1.66849i | −3.15408 | + | 3.87966i | 0 | −5.93757 | − | 5.93757i | −1.94768 | − | 3.82253i | 0 | −13.2607 | − | 7.69015i | ||||||
37.10 | 0.558161 | + | 3.52409i | 0 | −8.30342 | + | 2.69795i | 0.337675 | − | 4.98858i | 0 | −8.56695 | − | 8.56695i | −7.66306 | − | 15.0396i | 0 | 17.7687 | − | 1.59443i | ||||||
See all 80 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
25.f | odd | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.3.r.b | 80 | |
3.b | odd | 2 | 1 | 75.3.k.a | ✓ | 80 | |
15.d | odd | 2 | 1 | 375.3.k.a | 80 | ||
15.e | even | 4 | 1 | 375.3.k.b | 80 | ||
15.e | even | 4 | 1 | 375.3.k.c | 80 | ||
25.f | odd | 20 | 1 | inner | 225.3.r.b | 80 | |
75.h | odd | 10 | 1 | 375.3.k.b | 80 | ||
75.j | odd | 10 | 1 | 375.3.k.c | 80 | ||
75.l | even | 20 | 1 | 75.3.k.a | ✓ | 80 | |
75.l | even | 20 | 1 | 375.3.k.a | 80 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
75.3.k.a | ✓ | 80 | 3.b | odd | 2 | 1 | |
75.3.k.a | ✓ | 80 | 75.l | even | 20 | 1 | |
225.3.r.b | 80 | 1.a | even | 1 | 1 | trivial | |
225.3.r.b | 80 | 25.f | odd | 20 | 1 | inner | |
375.3.k.a | 80 | 15.d | odd | 2 | 1 | ||
375.3.k.a | 80 | 75.l | even | 20 | 1 | ||
375.3.k.b | 80 | 15.e | even | 4 | 1 | ||
375.3.k.b | 80 | 75.h | odd | 10 | 1 | ||
375.3.k.c | 80 | 15.e | even | 4 | 1 | ||
375.3.k.c | 80 | 75.j | odd | 10 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{80} + 4 T_{2}^{79} + 8 T_{2}^{78} - 4 T_{2}^{77} - 348 T_{2}^{76} - 580 T_{2}^{75} + \cdots + 87\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\).