Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,5,Mod(119,177)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.119");
S:= CuspForms(chi, 5);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.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: | \(18.2964834658\) |
Analytic rank: | \(0\) |
Dimension: | \(78\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
119.1 | − | 7.85862i | 1.36439 | + | 8.89598i | −45.7580 | 33.8616i | 69.9102 | − | 10.7222i | 13.8137 | 233.857i | −77.2769 | + | 24.2751i | 266.105 | |||||||||||
119.2 | − | 7.73459i | −8.52815 | − | 2.87588i | −43.8239 | − | 0.620129i | −22.2437 | + | 65.9618i | 71.8012 | 215.207i | 64.4587 | + | 49.0518i | −4.79645 | ||||||||||
119.3 | − | 7.49534i | −6.13251 | + | 6.58728i | −40.1801 | − | 24.9668i | 49.3739 | + | 45.9653i | −52.3496 | 181.238i | −5.78454 | − | 80.7932i | −187.134 | ||||||||||
119.4 | − | 7.42214i | 2.94091 | − | 8.50594i | −39.0882 | 22.5647i | −63.1323 | − | 21.8278i | 11.4871 | 171.364i | −63.7021 | − | 50.0304i | 167.478 | |||||||||||
119.5 | − | 7.37175i | 8.47141 | − | 3.03895i | −38.3427 | − | 40.5786i | −22.4024 | − | 62.4491i | 39.1148 | 164.705i | 62.5296 | − | 51.4883i | −299.136 | ||||||||||
119.6 | − | 7.08630i | −1.58963 | − | 8.85850i | −34.2156 | − | 8.65783i | −62.7740 | + | 11.2646i | −70.0243 | 129.081i | −75.9462 | + | 28.1634i | −61.3520 | ||||||||||
119.7 | − | 6.87418i | 8.98827 | + | 0.459381i | −31.2543 | 23.0718i | 3.15787 | − | 61.7870i | 23.0938 | 104.861i | 80.5779 | + | 8.25808i | 158.600 | |||||||||||
119.8 | − | 6.80672i | 7.51401 | + | 4.95376i | −30.3314 | − | 9.67205i | 33.7188 | − | 51.1457i | −60.1761 | 97.5501i | 31.9206 | + | 74.4451i | −65.8349 | ||||||||||
119.9 | − | 6.47467i | −8.82514 | − | 1.76549i | −25.9214 | 26.7513i | −11.4310 | + | 57.1399i | −48.5383 | 64.2378i | 74.7661 | + | 31.1614i | 173.206 | |||||||||||
119.10 | − | 6.42743i | 2.05646 | + | 8.76190i | −25.3118 | − | 39.0573i | 56.3165 | − | 13.2178i | 65.2885 | 59.8512i | −72.5419 | + | 36.0371i | −251.038 | ||||||||||
119.11 | − | 6.25305i | −4.60229 | − | 7.73427i | −23.1007 | − | 36.4421i | −48.3628 | + | 28.7783i | 14.4138 | 44.4009i | −38.6379 | + | 71.1907i | −227.875 | ||||||||||
119.12 | − | 5.88235i | −8.00524 | + | 4.11293i | −18.6020 | − | 15.2405i | 24.1937 | + | 47.0896i | 49.3247 | 15.3061i | 47.1676 | − | 65.8500i | −89.6502 | ||||||||||
119.13 | − | 5.74797i | −1.10979 | + | 8.93131i | −17.0392 | 1.93242i | 51.3369 | + | 6.37903i | −31.0983 | 5.97306i | −78.5367 | − | 19.8237i | 11.1075 | |||||||||||
119.14 | − | 5.57893i | −6.24762 | + | 6.47821i | −15.1244 | 38.3444i | 36.1415 | + | 34.8550i | 12.9947 | − | 4.88479i | −2.93443 | − | 80.9468i | 213.920 | ||||||||||
119.15 | − | 5.52059i | 4.22607 | + | 7.94609i | −14.4769 | 14.8774i | 43.8671 | − | 23.3304i | 83.7732 | − | 8.40835i | −45.2807 | + | 67.1614i | 82.1318 | ||||||||||
119.16 | − | 5.41049i | 7.81854 | − | 4.45762i | −13.2734 | 41.9044i | −24.1179 | − | 42.3021i | −64.0047 | − | 14.7525i | 41.2592 | − | 69.7042i | 226.723 | ||||||||||
119.17 | − | 5.25412i | 4.74448 | − | 7.64787i | −11.6058 | − | 7.04330i | −40.1829 | − | 24.9281i | 35.8552 | − | 23.0875i | −35.9798 | − | 72.5703i | −37.0064 | |||||||||
119.18 | − | 5.04906i | −3.85410 | − | 8.13301i | −9.49305 | 33.9624i | −41.0641 | + | 19.4596i | 88.4932 | − | 32.8540i | −51.2919 | + | 62.6909i | 171.478 | ||||||||||
119.19 | − | 4.62880i | 7.14239 | − | 5.47598i | −5.42582 | − | 33.7032i | −25.3472 | − | 33.0607i | −90.1786 | − | 48.9458i | 21.0273 | − | 78.2231i | −156.006 | |||||||||
119.20 | − | 4.27932i | −6.98370 | − | 5.67697i | −2.31259 | − | 6.15778i | −24.2936 | + | 29.8855i | −22.1503 | − | 58.5728i | 16.5440 | + | 79.2925i | −26.3511 | |||||||||
See all 78 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 177.5.b.a | ✓ | 78 |
3.b | odd | 2 | 1 | inner | 177.5.b.a | ✓ | 78 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.5.b.a | ✓ | 78 | 1.a | even | 1 | 1 | trivial |
177.5.b.a | ✓ | 78 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(177, [\chi])\).