Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,4,Mod(2,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(58))
chi = DirichletCharacter(H, H._module([29, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.2");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.f (of order \(58\), degree \(28\), 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: | \(10.4433380710\) |
Analytic rank: | \(0\) |
Dimension: | \(1624\) |
Relative dimension: | \(58\) over \(\Q(\zeta_{58})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{58}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2.1 | −0.301320 | + | 5.55751i | 4.73924 | − | 2.13064i | −22.8421 | − | 2.48423i | −5.05728 | + | 15.0095i | 10.4130 | + | 26.9804i | −9.02219 | − | 13.3067i | 13.4855 | − | 82.2577i | 17.9208 | − | 20.1952i | −81.8915 | − | 32.6285i |
2.2 | −0.285117 | + | 5.25867i | 5.18186 | + | 0.385093i | −19.6192 | − | 2.13372i | 6.02049 | − | 17.8682i | −3.50252 | + | 27.1399i | 18.4360 | + | 27.1910i | 9.99825 | − | 60.9866i | 26.7034 | + | 3.99100i | 92.2463 | + | 36.7543i |
2.3 | −0.282358 | + | 5.20779i | −0.412985 | + | 5.17971i | −19.0882 | − | 2.07597i | −2.66012 | + | 7.89495i | −26.8583 | − | 3.61327i | 2.27648 | + | 3.35756i | 9.45083 | − | 57.6475i | −26.6589 | − | 4.27829i | −40.3641 | − | 16.0825i |
2.4 | −0.281900 | + | 5.19934i | −5.12170 | + | 0.876440i | −19.0005 | − | 2.06643i | 2.67889 | − | 7.95067i | −3.11310 | − | 26.8765i | 1.88189 | + | 2.77557i | 9.36119 | − | 57.1007i | 25.4637 | − | 8.97773i | 40.5830 | + | 16.1698i |
2.5 | −0.273845 | + | 5.05077i | −1.98491 | − | 4.80210i | −17.4822 | − | 1.90130i | −0.350795 | + | 1.04112i | 24.7978 | − | 8.71029i | −5.00566 | − | 7.38279i | 7.84382 | − | 47.8452i | −19.1203 | + | 19.0635i | −5.16240 | − | 2.05689i |
2.6 | −0.254048 | + | 4.68564i | 1.63609 | + | 4.93186i | −13.9375 | − | 1.51580i | 5.44329 | − | 16.1551i | −23.5245 | + | 6.41321i | −18.4501 | − | 27.2119i | 4.56997 | − | 27.8756i | −21.6464 | + | 16.1380i | 74.3141 | + | 29.6094i |
2.7 | −0.247929 | + | 4.57278i | 0.343239 | − | 5.18480i | −12.8957 | − | 1.40250i | 2.38761 | − | 7.08619i | 23.6239 | + | 2.85502i | 6.25613 | + | 9.22711i | 3.68350 | − | 22.4684i | −26.7644 | − | 3.55926i | 31.8116 | + | 12.6749i |
2.8 | −0.238865 | + | 4.40561i | −4.39778 | − | 2.76759i | −11.3992 | − | 1.23974i | −6.88505 | + | 20.4341i | 13.2434 | − | 18.7138i | 12.3388 | + | 18.1984i | 2.47433 | − | 15.0928i | 11.6809 | + | 24.3425i | −88.3800 | − | 35.2138i |
2.9 | −0.216913 | + | 4.00073i | 4.38754 | − | 2.78379i | −8.00566 | − | 0.870667i | 2.33628 | − | 6.93383i | 10.1855 | + | 18.1572i | −13.7785 | − | 20.3217i | 0.0342646 | − | 0.209005i | 11.5010 | − | 24.4280i | 27.2336 | + | 10.8508i |
2.10 | −0.214445 | + | 3.95521i | 4.30093 | + | 2.91582i | −7.64463 | − | 0.831403i | −4.87721 | + | 14.4750i | −12.4550 | + | 16.3858i | 8.97536 | + | 13.2377i | −0.198845 | + | 1.21290i | 9.99602 | + | 25.0815i | −56.2060 | − | 22.3945i |
2.11 | −0.212337 | + | 3.91633i | 4.58888 | + | 2.43766i | −7.33945 | − | 0.798213i | −0.421405 | + | 1.25069i | −10.5211 | + | 17.4540i | −5.56806 | − | 8.21227i | −0.391673 | + | 2.38910i | 15.1156 | + | 22.3723i | −4.80862 | − | 1.91593i |
2.12 | −0.206723 | + | 3.81279i | −4.04421 | + | 3.26257i | −6.54149 | − | 0.711430i | −3.85784 | + | 11.4497i | −11.6035 | − | 16.0941i | −12.2513 | − | 18.0693i | −0.877159 | + | 5.35043i | 5.71121 | − | 26.3891i | −42.8576 | − | 17.0760i |
2.13 | −0.194915 | + | 3.59500i | 3.72957 | − | 3.61806i | −4.93296 | − | 0.536492i | −1.29349 | + | 3.83895i | 12.2800 | + | 14.1130i | 12.2031 | + | 17.9982i | −1.76949 | + | 10.7934i | 0.819319 | − | 26.9876i | −13.5489 | − | 5.39838i |
2.14 | −0.191601 | + | 3.53386i | −0.887175 | + | 5.11986i | −4.49838 | − | 0.489228i | 2.01755 | − | 5.98787i | −17.9229 | − | 4.11612i | 16.3407 | + | 24.1007i | −1.98968 | + | 12.1365i | −25.4258 | − | 9.08442i | 20.7738 | + | 8.27702i |
2.15 | −0.181075 | + | 3.33974i | −4.07453 | − | 3.22462i | −3.16798 | − | 0.344538i | 5.90079 | − | 17.5129i | 11.5072 | − | 13.0240i | −10.9419 | − | 16.1380i | −2.60452 | + | 15.8869i | 6.20361 | + | 26.2777i | 57.4202 | + | 22.8783i |
2.16 | −0.176293 | + | 3.25153i | −5.15278 | − | 0.670007i | −2.58825 | − | 0.281489i | −0.0532670 | + | 0.158091i | 3.08694 | − | 16.6363i | 11.3529 | + | 16.7443i | −2.84293 | + | 17.3411i | 26.1022 | + | 6.90479i | −0.504646 | − | 0.201069i |
2.17 | −0.150879 | + | 2.78279i | −4.15765 | + | 3.11672i | 0.231945 | + | 0.0252256i | 4.42533 | − | 13.1339i | −8.04587 | − | 12.0401i | −0.713636 | − | 1.05253i | −3.71213 | + | 22.6430i | 7.57215 | − | 25.9165i | 35.8813 | + | 14.2964i |
2.18 | −0.142776 | + | 2.63336i | 0.601836 | − | 5.16118i | 1.03892 | + | 0.112990i | −5.43994 | + | 16.1452i | 13.5053 | + | 2.32174i | −11.5153 | − | 16.9838i | −3.85912 | + | 23.5396i | −26.2756 | − | 6.21237i | −41.7393 | − | 16.6305i |
2.19 | −0.123545 | + | 2.27865i | −4.04503 | − | 3.26155i | 2.77611 | + | 0.301920i | −0.776377 | + | 2.30421i | 7.93169 | − | 8.81427i | −10.4076 | − | 15.3501i | −3.98444 | + | 24.3040i | 5.72454 | + | 26.3862i | −5.15457 | − | 2.05377i |
2.20 | −0.100013 | + | 1.84463i | 2.00907 | − | 4.79204i | 4.56043 | + | 0.495977i | 6.12126 | − | 18.1673i | 8.63862 | + | 4.18527i | 1.32430 | + | 1.95320i | −3.76193 | + | 22.9468i | −18.9273 | − | 19.2551i | 32.8997 | + | 13.1084i |
See next 80 embeddings (of 1624 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
59.d | odd | 58 | 1 | inner |
177.f | even | 58 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.4.f.a | ✓ | 1624 |
3.b | odd | 2 | 1 | inner | 177.4.f.a | ✓ | 1624 |
59.d | odd | 58 | 1 | inner | 177.4.f.a | ✓ | 1624 |
177.f | even | 58 | 1 | inner | 177.4.f.a | ✓ | 1624 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.4.f.a | ✓ | 1624 | 1.a | even | 1 | 1 | trivial |
177.4.f.a | ✓ | 1624 | 3.b | odd | 2 | 1 | inner |
177.4.f.a | ✓ | 1624 | 59.d | odd | 58 | 1 | inner |
177.4.f.a | ✓ | 1624 | 177.f | even | 58 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(177, [\chi])\).