Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,2,Mod(58,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(20))
chi = DirichletCharacter(H, H._module([0, 15, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("465.58");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 465.bk (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: | \(3.71304369399\) |
Analytic rank: | \(0\) |
Dimension: | \(256\) |
Relative dimension: | \(32\) over \(\Q(\zeta_{20})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
58.1 | −2.45364 | − | 1.25019i | 0.453990 | + | 0.891007i | 3.28181 | + | 4.51702i | 2.08158 | − | 0.816717i | − | 2.75379i | 0.0978996 | − | 0.618114i | −1.54366 | − | 9.74627i | −0.587785 | + | 0.809017i | −6.12850 | − | 0.598446i | |
58.2 | −2.40980 | − | 1.22786i | −0.453990 | − | 0.891007i | 3.12394 | + | 4.29974i | −1.85985 | − | 1.24136i | 2.70458i | −0.0505855 | + | 0.319384i | −1.40245 | − | 8.85470i | −0.587785 | + | 0.809017i | 2.95765 | + | 5.27505i | ||
58.3 | −2.12936 | − | 1.08496i | −0.453990 | − | 0.891007i | 2.18147 | + | 3.00253i | 2.21277 | + | 0.321924i | 2.38984i | 0.697204 | − | 4.40197i | −0.639786 | − | 4.03945i | −0.587785 | + | 0.809017i | −4.36252 | − | 3.08627i | ||
58.4 | −1.93862 | − | 0.987776i | 0.453990 | + | 0.891007i | 1.60697 | + | 2.21181i | 0.432638 | + | 2.19382i | − | 2.17576i | 0.0663490 | − | 0.418911i | −0.249810 | − | 1.57724i | −0.587785 | + | 0.809017i | 1.32828 | − | 4.68032i | |
58.5 | −1.81575 | − | 0.925171i | 0.453990 | + | 0.891007i | 1.26544 | + | 1.74172i | −1.15976 | − | 1.91179i | − | 2.03786i | 0.588758 | − | 3.71727i | −0.0487400 | − | 0.307732i | −0.587785 | + | 0.809017i | 0.337108 | + | 4.54432i | |
58.6 | −1.78720 | − | 0.910624i | 0.453990 | + | 0.891007i | 1.18928 | + | 1.63690i | −2.19888 | + | 0.406139i | − | 2.00582i | −0.793082 | + | 5.00732i | −0.00731841 | − | 0.0462066i | −0.587785 | + | 0.809017i | 4.29967 | + | 1.27650i | |
58.7 | −1.66600 | − | 0.848869i | −0.453990 | − | 0.891007i | 0.879406 | + | 1.21040i | −1.55591 | + | 1.60598i | 1.86980i | 0.0199774 | − | 0.126132i | 0.147381 | + | 0.930527i | −0.587785 | + | 0.809017i | 3.95540 | − | 1.35479i | ||
58.8 | −1.39268 | − | 0.709604i | −0.453990 | − | 0.891007i | 0.260440 | + | 0.358465i | −2.14105 | − | 0.644906i | 1.56304i | −0.0898134 | + | 0.567059i | 0.380685 | + | 2.40355i | −0.587785 | + | 0.809017i | 2.52416 | + | 2.41744i | ||
58.9 | −1.37147 | − | 0.698799i | 0.453990 | + | 0.891007i | 0.217040 | + | 0.298729i | 1.93562 | − | 1.11953i | − | 1.53924i | −0.281754 | + | 1.77893i | 0.392668 | + | 2.47921i | −0.587785 | + | 0.809017i | −3.43698 | + | 0.182796i | |
58.10 | −1.22557 | − | 0.624460i | −0.453990 | − | 0.891007i | −0.0634957 | − | 0.0873943i | 0.779662 | + | 2.09574i | 1.37549i | 0.140636 | − | 0.887942i | 0.453593 | + | 2.86387i | −0.587785 | + | 0.809017i | 0.353173 | − | 3.05535i | ||
58.11 | −0.857660 | − | 0.437000i | 0.453990 | + | 0.891007i | −0.630958 | − | 0.868439i | −2.13368 | + | 0.668885i | − | 0.962575i | 0.367045 | − | 2.31743i | 0.462800 | + | 2.92200i | −0.587785 | + | 0.809017i | 2.12228 | + | 0.358742i | |
58.12 | −0.638216 | − | 0.325188i | −0.453990 | − | 0.891007i | −0.873997 | − | 1.20295i | 2.17673 | + | 0.511700i | 0.716287i | −0.378452 | + | 2.38945i | 0.390718 | + | 2.46689i | −0.587785 | + | 0.809017i | −1.22283 | − | 1.03442i | ||
58.13 | −0.543929 | − | 0.277146i | −0.453990 | − | 0.891007i | −0.956521 | − | 1.31654i | −1.07645 | − | 1.95991i | 0.610466i | −0.750409 | + | 4.73789i | 0.346403 | + | 2.18710i | −0.587785 | + | 0.809017i | 0.0423294 | + | 1.36439i | ||
58.14 | −0.482709 | − | 0.245953i | 0.453990 | + | 0.891007i | −1.00306 | − | 1.38059i | 0.0551941 | − | 2.23539i | − | 0.541757i | −0.0625121 | + | 0.394686i | 0.314124 | + | 1.98330i | −0.587785 | + | 0.809017i | −0.576442 | + | 1.06547i | |
58.15 | −0.261959 | − | 0.133475i | 0.453990 | + | 0.891007i | −1.12476 | − | 1.54810i | 1.67870 | + | 1.47715i | − | 0.294004i | 0.701202 | − | 4.42721i | 0.179994 | + | 1.13644i | −0.587785 | + | 0.809017i | −0.242587 | − | 0.611018i | |
58.16 | −0.225656 | − | 0.114978i | 0.453990 | + | 0.891007i | −1.13787 | − | 1.56614i | −0.428506 | + | 2.19463i | − | 0.253260i | −0.172898 | + | 1.09164i | 0.155933 | + | 0.984523i | −0.587785 | + | 0.809017i | 0.349028 | − | 0.445962i | |
58.17 | −0.159639 | − | 0.0813400i | −0.453990 | − | 0.891007i | −1.15670 | − | 1.59206i | 1.27508 | − | 1.83689i | 0.179167i | 0.490854 | − | 3.09913i | 0.111212 | + | 0.702163i | −0.587785 | + | 0.809017i | −0.352964 | + | 0.189525i | ||
58.18 | −0.0651005 | − | 0.0331704i | −0.453990 | − | 0.891007i | −1.17243 | − | 1.61372i | −2.08874 | + | 0.798232i | 0.0730641i | 0.586605 | − | 3.70368i | 0.0456579 | + | 0.288273i | −0.587785 | + | 0.809017i | 0.162456 | + | 0.0173189i | ||
58.19 | 0.398088 | + | 0.202836i | 0.453990 | + | 0.891007i | −1.05824 | − | 1.45654i | −1.96506 | − | 1.06701i | 0.446785i | −0.584949 | + | 3.69322i | −0.265618 | − | 1.67705i | −0.587785 | + | 0.809017i | −0.565840 | − | 0.823352i | ||
58.20 | 0.480488 | + | 0.244821i | 0.453990 | + | 0.891007i | −1.00464 | − | 1.38277i | 2.17920 | + | 0.501102i | 0.539264i | −0.230590 | + | 1.45589i | −0.312906 | − | 1.97561i | −0.587785 | + | 0.809017i | 0.924397 | + | 0.774286i | ||
See next 80 embeddings (of 256 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
31.f | odd | 10 | 1 | inner |
155.r | even | 20 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 465.2.bk.a | ✓ | 256 |
5.c | odd | 4 | 1 | inner | 465.2.bk.a | ✓ | 256 |
31.f | odd | 10 | 1 | inner | 465.2.bk.a | ✓ | 256 |
155.r | even | 20 | 1 | inner | 465.2.bk.a | ✓ | 256 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
465.2.bk.a | ✓ | 256 | 1.a | even | 1 | 1 | trivial |
465.2.bk.a | ✓ | 256 | 5.c | odd | 4 | 1 | inner |
465.2.bk.a | ✓ | 256 | 31.f | odd | 10 | 1 | inner |
465.2.bk.a | ✓ | 256 | 155.r | even | 20 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(465, [\chi])\).