Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [671,2,Mod(353,671)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(671, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("671.353");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 671 = 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 671.o (of order \(6\), degree \(2\), 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: | \(5.35796197563\) |
Analytic rank: | \(0\) |
Dimension: | \(100\) |
Relative dimension: | \(50\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
353.1 | −2.28333 | − | 1.31828i | −2.61743 | 2.47573 | + | 4.28809i | 0.483876 | − | 0.838098i | 5.97645 | + | 3.45051i | −2.89686 | − | 1.67250i | − | 7.78170i | 3.85093 | −2.20970 | + | 1.27577i | |||||
353.2 | −2.25208 | − | 1.30024i | −1.39379 | 2.38123 | + | 4.12442i | −1.39465 | + | 2.41560i | 3.13893 | + | 1.81226i | 2.34580 | + | 1.35435i | − | 7.18372i | −1.05734 | 6.28170 | − | 3.62674i | |||||
353.3 | −2.16485 | − | 1.24988i | −0.750612 | 2.12440 | + | 3.67956i | 1.39042 | − | 2.40828i | 1.62496 | + | 0.938174i | 1.80357 | + | 1.04129i | − | 5.62144i | −2.43658 | −6.02013 | + | 3.47572i | |||||
353.4 | −2.04058 | − | 1.17813i | 1.10807 | 1.77599 | + | 3.07611i | −0.773884 | + | 1.34041i | −2.26110 | − | 1.30545i | 2.33677 | + | 1.34914i | − | 3.65688i | −1.77219 | 3.15835 | − | 1.82348i | |||||
353.5 | −1.95214 | − | 1.12707i | 3.30065 | 1.54058 | + | 2.66836i | −1.38341 | + | 2.39614i | −6.44334 | − | 3.72006i | 2.30926 | + | 1.33325i | − | 2.43708i | 7.89427 | 5.40124 | − | 3.11841i | |||||
353.6 | −1.94485 | − | 1.12286i | 2.13240 | 1.52163 | + | 2.63554i | −1.62555 | + | 2.81554i | −4.14721 | − | 2.39439i | −3.97334 | − | 2.29401i | − | 2.34287i | 1.54715 | 6.32292 | − | 3.65054i | |||||
353.7 | −1.93983 | − | 1.11996i | −0.234803 | 1.50863 | + | 2.61302i | 0.0821411 | − | 0.142272i | 0.455478 | + | 0.262970i | −2.78442 | − | 1.60758i | − | 2.27859i | −2.94487 | −0.318680 | + | 0.183990i | |||||
353.8 | −1.81114 | − | 1.04566i | −2.97181 | 1.18683 | + | 2.05564i | −1.33911 | + | 2.31941i | 5.38237 | + | 3.10751i | 3.08940 | + | 1.78367i | − | 0.781432i | 5.83165 | 4.85066 | − | 2.80053i | |||||
353.9 | −1.66014 | − | 0.958485i | −2.72985 | 0.837387 | + | 1.45040i | 1.58254 | − | 2.74105i | 4.53195 | + | 2.61652i | 2.39252 | + | 1.38132i | 0.623448i | 4.45211 | −5.25450 | + | 3.03369i | ||||||
353.10 | −1.61255 | − | 0.931003i | 2.83991 | 0.733534 | + | 1.27052i | 1.48075 | − | 2.56473i | −4.57948 | − | 2.64396i | −1.11196 | − | 0.641989i | 0.992322i | 5.06507 | −4.77554 | + | 2.75716i | ||||||
353.11 | −1.50557 | − | 0.869242i | 0.472199 | 0.511163 | + | 0.885361i | −0.131210 | + | 0.227263i | −0.710929 | − | 0.410455i | 3.18489 | + | 1.83880i | 1.69967i | −2.77703 | 0.395093 | − | 0.228107i | ||||||
353.12 | −1.48575 | − | 0.857801i | 1.46323 | 0.471645 | + | 0.816913i | 0.492748 | − | 0.853464i | −2.17399 | − | 1.25516i | −2.66588 | − | 1.53914i | 1.81289i | −0.858970 | −1.46420 | + | 0.845359i | ||||||
353.13 | −1.47682 | − | 0.852642i | −1.17833 | 0.453995 | + | 0.786343i | −0.185718 | + | 0.321673i | 1.74018 | + | 1.00469i | 0.148482 | + | 0.0857259i | 1.86218i | −1.61154 | 0.548544 | − | 0.316702i | ||||||
353.14 | −1.27894 | − | 0.738395i | −2.99796 | 0.0904543 | + | 0.156671i | 0.951983 | − | 1.64888i | 3.83420 | + | 2.21368i | −1.98281 | − | 1.14478i | 2.68642i | 5.98776 | −2.43505 | + | 1.40588i | ||||||
353.15 | −1.21522 | − | 0.701610i | 0.665795 | −0.0154872 | − | 0.0268246i | −1.61066 | + | 2.78975i | −0.809089 | − | 0.467128i | 0.374004 | + | 0.215931i | 2.84990i | −2.55672 | 3.91463 | − | 2.26011i | ||||||
353.16 | −1.00471 | − | 0.580068i | 0.804932 | −0.327042 | − | 0.566454i | 1.36581 | − | 2.36566i | −0.808721 | − | 0.466915i | −0.143518 | − | 0.0828603i | 3.07910i | −2.35209 | −2.74448 | + | 1.58453i | ||||||
353.17 | −0.948162 | − | 0.547422i | 2.71191 | −0.400659 | − | 0.693961i | −0.354654 | + | 0.614279i | −2.57133 | − | 1.48456i | 2.06456 | + | 1.19197i | 3.06700i | 4.35444 | 0.672540 | − | 0.388291i | ||||||
353.18 | −0.697945 | − | 0.402959i | 1.97028 | −0.675248 | − | 1.16956i | −1.47722 | + | 2.55862i | −1.37515 | − | 0.793941i | −1.60723 | − | 0.927937i | 2.70022i | 0.881997 | 2.06204 | − | 1.19052i | ||||||
353.19 | −0.582930 | − | 0.336555i | −2.24487 | −0.773462 | − | 1.33968i | 0.278495 | − | 0.482368i | 1.30860 | + | 0.755521i | 1.53539 | + | 0.886457i | 2.38747i | 2.03943 | −0.324687 | + | 0.187458i | ||||||
353.20 | −0.529811 | − | 0.305887i | −1.44271 | −0.812867 | − | 1.40793i | −0.00570608 | + | 0.00988322i | 0.764366 | + | 0.441307i | −1.63348 | − | 0.943088i | 2.21813i | −0.918576 | 0.00604629 | − | 0.00349083i | ||||||
See all 100 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
61.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 671.2.o.a | ✓ | 100 |
61.f | even | 6 | 1 | inner | 671.2.o.a | ✓ | 100 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
671.2.o.a | ✓ | 100 | 1.a | even | 1 | 1 | trivial |
671.2.o.a | ✓ | 100 | 61.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).