Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [833,2,Mod(48,833)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([8, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("833.48");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 833 = 7^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 833.t (of order \(16\), 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: | \(6.65153848837\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(9\) over \(\Q(\zeta_{16})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{16}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
48.1 | −2.56233 | + | 1.06135i | −1.71514 | − | 0.341162i | 4.02484 | − | 4.02484i | −0.155648 | − | 0.232944i | 4.75684 | − | 0.946194i | 0 | −3.91849 | + | 9.46008i | 0.0536674 | + | 0.0222297i | 0.646056 | + | 0.431681i | ||
48.2 | −2.18495 | + | 0.905036i | 3.23570 | + | 0.643620i | 2.54070 | − | 2.54070i | −1.04445 | − | 1.56313i | −7.65233 | + | 1.52214i | 0 | −1.44181 | + | 3.48083i | 7.28385 | + | 3.01707i | 3.69676 | + | 2.47009i | ||
48.3 | −1.52417 | + | 0.631334i | 0.480778 | + | 0.0956326i | 0.510311 | − | 0.510311i | 1.43429 | + | 2.14657i | −0.793165 | + | 0.157770i | 0 | 0.807041 | − | 1.94837i | −2.54964 | − | 1.05609i | −3.54131 | − | 2.36623i | ||
48.4 | −0.985259 | + | 0.408108i | −2.83391 | − | 0.563700i | −0.610030 | + | 0.610030i | 0.380055 | + | 0.568792i | 3.02219 | − | 0.601151i | 0 | 1.16830 | − | 2.82051i | 4.94167 | + | 2.04690i | −0.606580 | − | 0.405304i | ||
48.5 | −0.355236 | + | 0.147143i | 1.41465 | + | 0.281392i | −1.30967 | + | 1.30967i | −0.839778 | − | 1.25682i | −0.543939 | + | 0.108196i | 0 | 0.566820 | − | 1.36842i | −0.849583 | − | 0.351909i | 0.483251 | + | 0.322898i | ||
48.6 | 0.796325 | − | 0.329849i | −1.48546 | − | 0.295477i | −0.888880 | + | 0.888880i | −1.62687 | − | 2.43478i | −1.28037 | + | 0.254682i | 0 | −1.07434 | + | 2.59368i | −0.652344 | − | 0.270210i | −2.09862 | − | 1.40226i | ||
48.7 | 1.36887 | − | 0.567005i | 2.16558 | + | 0.430761i | 0.138099 | − | 0.138099i | 1.47955 | + | 2.21431i | 3.20864 | − | 0.638239i | 0 | −1.02327 | + | 2.47040i | 1.73255 | + | 0.717644i | 3.28084 | + | 2.19219i | ||
48.8 | 2.18677 | − | 0.905789i | −3.01102 | − | 0.598929i | 2.54729 | − | 2.54729i | 1.18717 | + | 1.77673i | −7.12690 | + | 1.41763i | 0 | 1.45144 | − | 3.50409i | 5.93588 | + | 2.45872i | 4.20540 | + | 2.80996i | ||
48.9 | 2.33610 | − | 0.967645i | 0.914736 | + | 0.181952i | 3.10682 | − | 3.10682i | −0.998100 | − | 1.49376i | 2.31298 | − | 0.460081i | 0 | 2.31626 | − | 5.59195i | −1.96800 | − | 0.815173i | −3.77709 | − | 2.52377i | ||
97.1 | −1.02069 | − | 2.46415i | 0.692606 | + | 1.03656i | −3.61604 | + | 3.61604i | 0.102295 | − | 0.514273i | 1.84730 | − | 2.76469i | 0 | 7.67299 | + | 3.17826i | 0.553301 | − | 1.33579i | −1.37166 | + | 0.272840i | ||
97.2 | −0.684321 | − | 1.65210i | −0.302826 | − | 0.453212i | −0.846919 | + | 0.846919i | −0.523006 | + | 2.62933i | −0.541520 | + | 0.810442i | 0 | −1.32544 | − | 0.549014i | 1.03435 | − | 2.49715i | 4.70181 | − | 0.935249i | ||
97.3 | −0.601251 | − | 1.45155i | −1.18492 | − | 1.77336i | −0.331273 | + | 0.331273i | 0.240255 | − | 1.20785i | −1.86168 | + | 2.78620i | 0 | −2.22306 | − | 0.920820i | −0.592708 | + | 1.43092i | −1.89770 | + | 0.377476i | ||
97.4 | −0.252613 | − | 0.609862i | 0.947770 | + | 1.41844i | 1.10610 | − | 1.10610i | 0.849136 | − | 4.26890i | 0.625632 | − | 0.936325i | 0 | −2.17370 | − | 0.900378i | 0.0343520 | − | 0.0829330i | −2.81794 | + | 0.560523i | ||
97.5 | 0.0230395 | + | 0.0556222i | 0.638303 | + | 0.955287i | 1.41165 | − | 1.41165i | −0.456570 | + | 2.29533i | −0.0384290 | + | 0.0575131i | 0 | 0.222287 | + | 0.0920744i | 0.642907 | − | 1.55211i | −0.138191 | + | 0.0274878i | ||
97.6 | 0.422335 | + | 1.01961i | −1.37804 | − | 2.06239i | 0.552980 | − | 0.552980i | 0.370260 | − | 1.86142i | 1.52083 | − | 2.27608i | 0 | 2.83658 | + | 1.17495i | −1.20639 | + | 2.91248i | 2.05430 | − | 0.408625i | ||
97.7 | 0.672106 | + | 1.62261i | 1.35675 | + | 2.03053i | −0.766915 | + | 0.766915i | 0.0982159 | − | 0.493765i | −2.38286 | + | 3.56621i | 0 | 1.48536 | + | 0.615258i | −1.13420 | + | 2.73821i | 0.867198 | − | 0.172496i | ||
97.8 | 0.768113 | + | 1.85439i | −0.823215 | − | 1.23203i | −1.43455 | + | 1.43455i | −0.743583 | + | 3.73825i | 1.65234 | − | 2.47290i | 0 | −0.0533216 | − | 0.0220865i | 0.307840 | − | 0.743193i | −7.50332 | + | 1.49250i | ||
97.9 | 1.05596 | + | 2.54931i | −0.346973 | − | 0.519282i | −3.96973 | + | 3.96973i | 0.635722 | − | 3.19599i | 0.957422 | − | 1.43288i | 0 | −9.21334 | − | 3.81629i | 0.998787 | − | 2.41129i | 8.81888 | − | 1.75418i | ||
146.1 | −1.02069 | + | 2.46415i | 0.692606 | − | 1.03656i | −3.61604 | − | 3.61604i | 0.102295 | + | 0.514273i | 1.84730 | + | 2.76469i | 0 | 7.67299 | − | 3.17826i | 0.553301 | + | 1.33579i | −1.37166 | − | 0.272840i | ||
146.2 | −0.684321 | + | 1.65210i | −0.302826 | + | 0.453212i | −0.846919 | − | 0.846919i | −0.523006 | − | 2.62933i | −0.541520 | − | 0.810442i | 0 | −1.32544 | + | 0.549014i | 1.03435 | + | 2.49715i | 4.70181 | + | 0.935249i | ||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
119.p | even | 16 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 833.2.t.a | ✓ | 72 |
7.b | odd | 2 | 1 | 833.2.t.d | yes | 72 | |
7.c | even | 3 | 2 | 833.2.bc.d | 144 | ||
7.d | odd | 6 | 2 | 833.2.bc.a | 144 | ||
17.e | odd | 16 | 1 | 833.2.t.d | yes | 72 | |
119.p | even | 16 | 1 | inner | 833.2.t.a | ✓ | 72 |
119.s | even | 48 | 2 | 833.2.bc.d | 144 | ||
119.t | odd | 48 | 2 | 833.2.bc.a | 144 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
833.2.t.a | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
833.2.t.a | ✓ | 72 | 119.p | even | 16 | 1 | inner |
833.2.t.d | yes | 72 | 7.b | odd | 2 | 1 | |
833.2.t.d | yes | 72 | 17.e | odd | 16 | 1 | |
833.2.bc.a | 144 | 7.d | odd | 6 | 2 | ||
833.2.bc.a | 144 | 119.t | odd | 48 | 2 | ||
833.2.bc.d | 144 | 7.c | even | 3 | 2 | ||
833.2.bc.d | 144 | 119.s | even | 48 | 2 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):
\( T_{2}^{72} + 8 T_{2}^{67} - 72 T_{2}^{65} + 5079 T_{2}^{64} - 576 T_{2}^{63} - 432 T_{2}^{62} + 880 T_{2}^{61} - 576 T_{2}^{60} + 41560 T_{2}^{59} + 36448 T_{2}^{58} - 110448 T_{2}^{57} + 7139489 T_{2}^{56} - 958024 T_{2}^{55} + \cdots + 16129 \) |
\( T_{3}^{72} + 8 T_{3}^{71} + 20 T_{3}^{70} - 8 T_{3}^{69} - 160 T_{3}^{68} - 624 T_{3}^{67} - 1288 T_{3}^{66} + 1728 T_{3}^{65} + 12832 T_{3}^{64} + 25176 T_{3}^{63} + 33236 T_{3}^{62} - 97792 T_{3}^{61} - 209976 T_{3}^{60} + \cdots + 358982369792 \) |