Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [208,4,Mod(77,208)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(208, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("208.77");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 208 = 2^{4} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 208.p (of order \(4\), 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: | \(12.2723972812\) |
Analytic rank: | \(0\) |
Dimension: | \(164\) |
Relative dimension: | \(82\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
77.1 | −2.82494 | + | 0.140455i | 6.40502 | + | 6.40502i | 7.96054 | − | 0.793553i | 2.11354 | + | 2.11354i | −18.9934 | − | 17.1942i | 17.9223 | −22.3766 | + | 3.35983i | 55.0486i | −6.26748 | − | 5.67377i | ||||
77.2 | −2.82451 | + | 0.148855i | −2.12524 | − | 2.12524i | 7.95568 | − | 0.840882i | −12.7715 | − | 12.7715i | 6.31910 | + | 5.68639i | 12.9494 | −22.3457 | + | 3.55932i | − | 17.9667i | 37.9744 | + | 34.1722i | |||
77.3 | −2.81839 | − | 0.238103i | −3.35598 | − | 3.35598i | 7.88661 | + | 1.34213i | 4.60987 | + | 4.60987i | 8.65939 | + | 10.2575i | −11.3328 | −21.9080 | − | 5.66047i | − | 4.47474i | −11.8948 | − | 14.0900i | |||
77.4 | −2.81645 | − | 0.260067i | 1.73668 | + | 1.73668i | 7.86473 | + | 1.46493i | −5.75097 | − | 5.75097i | −4.43961 | − | 5.34292i | −21.8456 | −21.7696 | − | 6.17126i | − | 20.9679i | 14.7016 | + | 17.6929i | |||
77.5 | −2.78961 | + | 0.466994i | −5.87109 | − | 5.87109i | 7.56383 | − | 2.60546i | 14.3059 | + | 14.3059i | 19.1198 | + | 13.6363i | 13.5187 | −19.8834 | + | 10.8005i | 41.9395i | −46.5885 | − | 33.2270i | ||||
77.6 | −2.77687 | + | 0.537582i | −0.574375 | − | 0.574375i | 7.42201 | − | 2.98559i | 1.31602 | + | 1.31602i | 1.90374 | + | 1.28619i | 32.3407 | −19.0050 | + | 12.2805i | − | 26.3402i | −4.36189 | − | 2.94695i | |||
77.7 | −2.77335 | + | 0.555438i | 2.56937 | + | 2.56937i | 7.38298 | − | 3.08085i | 7.06578 | + | 7.06578i | −8.55291 | − | 5.69865i | −19.8768 | −18.7644 | + | 12.6451i | − | 13.7966i | −23.5205 | − | 15.6713i | |||
77.8 | −2.62224 | − | 1.06013i | 3.73069 | + | 3.73069i | 5.75225 | + | 5.55982i | −8.80493 | − | 8.80493i | −5.82774 | − | 13.7378i | 9.35030 | −9.18965 | − | 20.6773i | 0.836117i | 13.7542 | + | 32.4230i | ||||
77.9 | −2.62201 | − | 1.06070i | 0.373419 | + | 0.373419i | 5.74983 | + | 5.56233i | 7.38008 | + | 7.38008i | −0.583021 | − | 1.37519i | −12.9160 | −9.17611 | − | 20.6833i | − | 26.7211i | −11.5226 | − | 27.1787i | |||
77.10 | −2.60879 | − | 1.09281i | −6.01467 | − | 6.01467i | 5.61154 | + | 5.70181i | −4.12290 | − | 4.12290i | 9.11812 | + | 22.2639i | −6.88029 | −8.40833 | − | 21.0071i | 45.3526i | 6.25022 | + | 15.2613i | ||||
77.11 | −2.60768 | + | 1.09546i | −4.82487 | − | 4.82487i | 5.59994 | − | 5.71320i | −5.24233 | − | 5.24233i | 17.8671 | + | 7.29625i | −32.3272 | −8.34425 | + | 21.0327i | 19.5588i | 19.4130 | + | 7.92753i | ||||
77.12 | −2.53500 | + | 1.25450i | 5.09937 | + | 5.09937i | 4.85245 | − | 6.36033i | −10.2683 | − | 10.2683i | −19.3241 | − | 6.52973i | −7.54188 | −4.32190 | + | 22.2108i | 25.0072i | 38.9116 | + | 13.1485i | ||||
77.13 | −2.47917 | − | 1.36152i | −4.76379 | − | 4.76379i | 4.29254 | + | 6.75086i | −0.130556 | − | 0.130556i | 5.32425 | + | 18.2962i | 15.3712 | −1.45052 | − | 22.5809i | 18.3873i | 0.145916 | + | 0.501425i | ||||
77.14 | −2.47587 | − | 1.36750i | 1.48929 | + | 1.48929i | 4.25989 | + | 6.77151i | 14.2987 | + | 14.2987i | −1.65068 | − | 5.72388i | 19.3394 | −1.28690 | − | 22.5908i | − | 22.5641i | −15.8483 | − | 54.9554i | |||
77.15 | −2.41290 | − | 1.47578i | 6.93624 | + | 6.93624i | 3.64414 | + | 7.12182i | 7.79313 | + | 7.79313i | −6.50005 | − | 26.9728i | −35.4139 | 1.71731 | − | 22.5622i | 69.2228i | −7.30306 | − | 30.3050i | ||||
77.16 | −2.41193 | + | 1.47737i | 3.87772 | + | 3.87772i | 3.63478 | − | 7.12659i | 12.4454 | + | 12.4454i | −15.0816 | − | 3.62397i | −4.13238 | 1.76175 | + | 22.5587i | 3.07347i | −48.4036 | − | 11.6309i | ||||
77.17 | −2.37565 | + | 1.53502i | −7.05438 | − | 7.05438i | 3.28740 | − | 7.29335i | −7.17035 | − | 7.17035i | 27.5874 | + | 5.93007i | 18.7503 | 3.38577 | + | 22.3727i | 72.5286i | 28.0409 | + | 6.02756i | ||||
77.18 | −2.22790 | + | 1.74254i | −2.07519 | − | 2.07519i | 1.92708 | − | 7.76443i | 1.67350 | + | 1.67350i | 8.23941 | + | 1.00720i | −4.91715 | 9.23652 | + | 20.6564i | − | 18.3872i | −6.64455 | − | 0.812245i | |||
77.19 | −2.18649 | + | 1.79423i | 0.279568 | + | 0.279568i | 1.56151 | − | 7.84613i | −9.66462 | − | 9.66462i | −1.11288 | − | 0.109666i | 2.26833 | 10.6635 | + | 19.9572i | − | 26.8437i | 38.4721 | + | 3.79113i | |||
77.20 | −2.05920 | − | 1.93899i | 4.45439 | + | 4.45439i | 0.480628 | + | 7.98555i | −2.81162 | − | 2.81162i | −0.535468 | − | 17.8095i | 15.8553 | 14.4942 | − | 17.3758i | 12.6832i | 0.337989 | + | 11.2414i | ||||
See next 80 embeddings (of 164 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
13.b | even | 2 | 1 | inner |
16.e | even | 4 | 1 | inner |
208.p | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 208.4.p.a | ✓ | 164 |
13.b | even | 2 | 1 | inner | 208.4.p.a | ✓ | 164 |
16.e | even | 4 | 1 | inner | 208.4.p.a | ✓ | 164 |
208.p | even | 4 | 1 | inner | 208.4.p.a | ✓ | 164 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
208.4.p.a | ✓ | 164 | 1.a | even | 1 | 1 | trivial |
208.4.p.a | ✓ | 164 | 13.b | even | 2 | 1 | inner |
208.4.p.a | ✓ | 164 | 16.e | even | 4 | 1 | inner |
208.4.p.a | ✓ | 164 | 208.p | even | 4 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(208, [\chi])\).