Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,2,Mod(8,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([2, 27]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.8");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 405.r (of order \(36\), degree \(12\), not 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: | \(3.23394128186\) |
Analytic rank: | \(0\) |
Dimension: | \(192\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{36})\) |
Twist minimal: | no (minimal twist has level 135) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-expansion
The algebraic \(q\)-expansion of this newform has not been computed, 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
8.1 | −2.05499 | + | 1.43892i | 0 | 1.46845 | − | 4.03452i | 1.82626 | + | 1.29027i | 0 | −1.14217 | + | 2.44940i | 1.48912 | + | 5.55747i | 0 | −5.60952 | − | 0.0236494i | ||||||
8.2 | −1.94894 | + | 1.36466i | 0 | 1.25203 | − | 3.43992i | −0.278496 | − | 2.21866i | 0 | −1.02475 | + | 2.19758i | 1.02263 | + | 3.81650i | 0 | 3.57049 | + | 3.94398i | ||||||
8.3 | −1.65044 | + | 1.15565i | 0 | 0.704386 | − | 1.93528i | 0.268922 | + | 2.21984i | 0 | 0.618828 | − | 1.32708i | 0.0310202 | + | 0.115769i | 0 | −3.00920 | − | 3.35293i | ||||||
8.4 | −1.23715 | + | 0.866263i | 0 | 0.0960923 | − | 0.264011i | 2.23168 | − | 0.140023i | 0 | 0.677826 | − | 1.45360i | −0.671958 | − | 2.50778i | 0 | −2.63963 | + | 2.10645i | ||||||
8.5 | −0.865920 | + | 0.606324i | 0 | −0.301851 | + | 0.829330i | −2.13636 | + | 0.660292i | 0 | −1.22161 | + | 2.61975i | −0.788655 | − | 2.94330i | 0 | 1.44956 | − | 1.86708i | ||||||
8.6 | −0.490069 | + | 0.343150i | 0 | −0.561625 | + | 1.54305i | 1.25284 | − | 1.85213i | 0 | −0.0636379 | + | 0.136472i | −0.563947 | − | 2.10468i | 0 | 0.0215816 | + | 1.33758i | ||||||
8.7 | −0.263290 | + | 0.184358i | 0 | −0.648706 | + | 1.78231i | −2.21531 | − | 0.303968i | 0 | 2.07991 | − | 4.46037i | −0.324162 | − | 1.20979i | 0 | 0.639309 | − | 0.328378i | ||||||
8.8 | −0.0928047 | + | 0.0649826i | 0 | −0.679650 | + | 1.86732i | −0.584024 | + | 2.15845i | 0 | 0.344848 | − | 0.739528i | −0.116914 | − | 0.436329i | 0 | −0.0860616 | − | 0.238266i | ||||||
8.9 | 0.377486 | − | 0.264318i | 0 | −0.611409 | + | 1.67983i | −0.538986 | − | 2.17014i | 0 | −1.52168 | + | 3.26325i | 0.451753 | + | 1.68596i | 0 | −0.777066 | − | 0.676732i | ||||||
8.10 | 0.807333 | − | 0.565301i | 0 | −0.351818 | + | 0.966613i | 2.04189 | + | 0.911409i | 0 | −0.275357 | + | 0.590505i | 0.772562 | + | 2.88324i | 0 | 2.16371 | − | 0.418474i | ||||||
8.11 | 1.08817 | − | 0.761943i | 0 | −0.0804885 | + | 0.221140i | −1.23426 | − | 1.86457i | 0 | 0.827388 | − | 1.77434i | 0.768546 | + | 2.86825i | 0 | −2.76377 | − | 1.08853i | ||||||
8.12 | 1.22359 | − | 0.856769i | 0 | 0.0790861 | − | 0.217287i | −2.10603 | + | 0.751431i | 0 | −2.03610 | + | 4.36642i | 0.683816 | + | 2.55204i | 0 | −1.93312 | + | 2.72382i | ||||||
8.13 | 1.42277 | − | 0.996232i | 0 | 0.347747 | − | 0.955427i | 0.893670 | + | 2.04972i | 0 | 0.429055 | − | 0.920112i | 0.442010 | + | 1.64960i | 0 | 3.31348 | + | 2.02597i | ||||||
8.14 | 1.66533 | − | 1.16608i | 0 | 0.729546 | − | 2.00441i | 1.48319 | − | 1.67336i | 0 | 1.13512 | − | 2.43427i | −0.0700073 | − | 0.261271i | 0 | 0.518731 | − | 4.51621i | ||||||
8.15 | 2.12047 | − | 1.48477i | 0 | 1.60780 | − | 4.41741i | 2.09416 | − | 0.783897i | 0 | −1.25154 | + | 2.68394i | −1.80956 | − | 6.75336i | 0 | 3.27669 | − | 4.77156i | ||||||
8.16 | 2.18018 | − | 1.52658i | 0 | 1.73870 | − | 4.77703i | −1.64880 | + | 1.51045i | 0 | 1.30062 | − | 2.78918i | −2.12414 | − | 7.92739i | 0 | −1.28885 | + | 5.81007i | ||||||
17.1 | −2.47401 | + | 1.15365i | 0 | 3.50423 | − | 4.17618i | 1.69173 | − | 1.46221i | 0 | −0.0671403 | − | 0.00587402i | −2.43862 | + | 9.10104i | 0 | −2.49848 | + | 5.56917i | ||||||
17.2 | −1.82102 | + | 0.849158i | 0 | 1.30949 | − | 1.56058i | 1.44997 | + | 1.70223i | 0 | 3.18459 | + | 0.278616i | −0.0193446 | + | 0.0721951i | 0 | −4.08589 | − | 1.86855i | ||||||
17.3 | −1.81626 | + | 0.846935i | 0 | 1.29592 | − | 1.54442i | −0.828498 | − | 2.07692i | 0 | −2.21198 | − | 0.193523i | −0.00834683 | + | 0.0311508i | 0 | 3.26378 | + | 3.07054i | ||||||
17.4 | −1.64734 | + | 0.768168i | 0 | 0.838077 | − | 0.998782i | −0.861745 | + | 2.06335i | 0 | −4.49995 | − | 0.393694i | 0.327512 | − | 1.22229i | 0 | −0.165408 | − | 4.06100i | ||||||
See next 80 embeddings (of 192 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
27.f | odd | 18 | 1 | inner |
135.q | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 405.2.r.a | 192 | |
3.b | odd | 2 | 1 | 135.2.q.a | ✓ | 192 | |
5.c | odd | 4 | 1 | inner | 405.2.r.a | 192 | |
15.d | odd | 2 | 1 | 675.2.ba.b | 192 | ||
15.e | even | 4 | 1 | 135.2.q.a | ✓ | 192 | |
15.e | even | 4 | 1 | 675.2.ba.b | 192 | ||
27.e | even | 9 | 1 | 135.2.q.a | ✓ | 192 | |
27.f | odd | 18 | 1 | inner | 405.2.r.a | 192 | |
135.p | even | 18 | 1 | 675.2.ba.b | 192 | ||
135.q | even | 36 | 1 | inner | 405.2.r.a | 192 | |
135.r | odd | 36 | 1 | 135.2.q.a | ✓ | 192 | |
135.r | odd | 36 | 1 | 675.2.ba.b | 192 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
135.2.q.a | ✓ | 192 | 3.b | odd | 2 | 1 | |
135.2.q.a | ✓ | 192 | 15.e | even | 4 | 1 | |
135.2.q.a | ✓ | 192 | 27.e | even | 9 | 1 | |
135.2.q.a | ✓ | 192 | 135.r | odd | 36 | 1 | |
405.2.r.a | 192 | 1.a | even | 1 | 1 | trivial | |
405.2.r.a | 192 | 5.c | odd | 4 | 1 | inner | |
405.2.r.a | 192 | 27.f | odd | 18 | 1 | inner | |
405.2.r.a | 192 | 135.q | even | 36 | 1 | inner | |
675.2.ba.b | 192 | 15.d | odd | 2 | 1 | ||
675.2.ba.b | 192 | 15.e | even | 4 | 1 | ||
675.2.ba.b | 192 | 135.p | even | 18 | 1 | ||
675.2.ba.b | 192 | 135.r | odd | 36 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(405, [\chi])\).