Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1205,2,Mod(724,1205)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1205, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1205.724");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1205 = 5 \cdot 241 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1205.b (of order \(2\), degree \(1\), 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: | \(9.62197344356\) |
Analytic rank: | \(0\) |
Dimension: | \(66\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
724.1 | − | 2.81437i | − | 0.650611i | −5.92068 | 2.22726 | + | 0.198227i | −1.83106 | − | 2.61686i | 11.0342i | 2.57671 | 0.557884 | − | 6.26834i | |||||||||||
724.2 | − | 2.73687i | 1.37229i | −5.49044 | −1.04813 | − | 1.97520i | 3.75578 | − | 0.639768i | 9.55288i | 1.11681 | −5.40587 | + | 2.86858i | ||||||||||||
724.3 | − | 2.70326i | 3.35528i | −5.30763 | 0.474898 | + | 2.18506i | 9.07021 | 3.43095i | 8.94141i | −8.25791 | 5.90678 | − | 1.28377i | |||||||||||||
724.4 | − | 2.60935i | − | 2.94245i | −4.80870 | −1.75698 | − | 1.38312i | −7.67788 | 2.28979i | 7.32887i | −5.65803 | −3.60904 | + | 4.58458i | ||||||||||||
724.5 | − | 2.55975i | − | 2.19931i | −4.55230 | −2.23588 | − | 0.0286488i | −5.62967 | − | 4.06530i | 6.53325i | −1.83695 | −0.0733336 | + | 5.72330i | |||||||||||
724.6 | − | 2.48606i | 2.54168i | −4.18051 | 2.19521 | − | 0.425495i | 6.31876 | − | 3.51286i | 5.42088i | −3.46011 | −1.05781 | − | 5.45743i | ||||||||||||
724.7 | − | 2.48219i | 1.89252i | −4.16124 | −1.31757 | + | 1.80665i | 4.69758 | − | 3.72710i | 5.36461i | −0.581628 | 4.48445 | + | 3.27046i | ||||||||||||
724.8 | − | 2.37391i | − | 0.703313i | −3.63543 | 1.41667 | − | 1.73005i | −1.66960 | 4.47866i | 3.88235i | 2.50535 | −4.10697 | − | 3.36303i | ||||||||||||
724.9 | − | 2.33352i | − | 1.59052i | −3.44531 | −0.227123 | + | 2.22450i | −3.71151 | − | 3.91727i | 3.37267i | 0.470249 | 5.19092 | + | 0.529996i | |||||||||||
724.10 | − | 2.31942i | 2.66924i | −3.37972 | 0.686453 | − | 2.12809i | 6.19110 | 0.0358936i | 3.20016i | −4.12486 | −4.93595 | − | 1.59218i | |||||||||||||
724.11 | − | 2.20997i | 2.26852i | −2.88397 | −2.23543 | − | 0.0532972i | 5.01336 | 0.372373i | 1.95355i | −2.14617 | −0.117785 | + | 4.94024i | |||||||||||||
724.12 | − | 2.06213i | − | 2.32228i | −2.25239 | 2.09119 | + | 0.791781i | −4.78886 | 3.49367i | 0.520471i | −2.39299 | 1.63276 | − | 4.31232i | ||||||||||||
724.13 | − | 1.90752i | − | 0.479938i | −1.63863 | −2.22437 | − | 0.228456i | −0.915491 | 1.00580i | − | 0.689327i | 2.76966 | −0.435785 | + | 4.24302i | |||||||||||
724.14 | − | 1.85162i | − | 0.305536i | −1.42851 | 2.21384 | − | 0.314518i | −0.565737 | − | 2.51443i | − | 1.05818i | 2.90665 | −0.582370 | − | 4.09920i | ||||||||||
724.15 | − | 1.85071i | − | 2.57343i | −1.42511 | 1.63741 | + | 1.52279i | −4.76266 | − | 3.52351i | − | 1.06395i | −3.62254 | 2.81824 | − | 3.03036i | ||||||||||
724.16 | − | 1.67685i | − | 3.17567i | −0.811815 | −1.72522 | + | 1.42254i | −5.32511 | 0.164609i | − | 1.99240i | −7.08488 | 2.38538 | + | 2.89293i | |||||||||||
724.17 | − | 1.64329i | 3.10101i | −0.700400 | 1.37323 | + | 1.76472i | 5.09585 | − | 0.459168i | − | 2.13562i | −6.61624 | 2.89995 | − | 2.25661i | |||||||||||
724.18 | − | 1.54552i | 0.0372080i | −0.388631 | −0.615725 | + | 2.14962i | 0.0575057 | 5.06287i | − | 2.49040i | 2.99862 | 3.32229 | + | 0.951615i | ||||||||||||
724.19 | − | 1.47177i | 1.07414i | −0.166115 | 1.15452 | − | 1.91496i | 1.58089 | − | 3.64779i | − | 2.69906i | 1.84623 | −2.81839 | − | 1.69919i | |||||||||||
724.20 | − | 1.38861i | 0.0366779i | 0.0717484 | 0.203412 | − | 2.22680i | 0.0509315 | 3.69082i | − | 2.87686i | 2.99865 | −3.09216 | − | 0.282461i | ||||||||||||
See all 66 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1205.2.b.d | ✓ | 66 |
5.b | even | 2 | 1 | inner | 1205.2.b.d | ✓ | 66 |
5.c | odd | 4 | 2 | 6025.2.a.q | 66 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1205.2.b.d | ✓ | 66 | 1.a | even | 1 | 1 | trivial |
1205.2.b.d | ✓ | 66 | 5.b | even | 2 | 1 | inner |
6025.2.a.q | 66 | 5.c | odd | 4 | 2 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 105 T_{2}^{64} + 5239 T_{2}^{62} + 165314 T_{2}^{60} + 3703934 T_{2}^{58} + 62721100 T_{2}^{56} + \cdots + 1841449 \) acting on \(S_{2}^{\mathrm{new}}(1205, [\chi])\).