Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [211,10,Mod(1,211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("211.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 211 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 211.a (trivial) |
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: | yes |
Analytic conductor: | \(108.672561431\) |
Analytic rank: | \(0\) |
Dimension: | \(82\) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 | −42.9522 | −272.921 | 1332.89 | 2616.34 | 11722.6 | −2274.16 | −35259.0 | 54803.0 | −112377. | ||||||||||||||||||
1.2 | −42.7495 | 173.421 | 1315.52 | −921.781 | −7413.66 | −9584.90 | −34350.2 | 10391.8 | 39405.7 | ||||||||||||||||||
1.3 | −41.5078 | 198.276 | 1210.90 | −1651.51 | −8229.98 | −7477.50 | −29009.7 | 19630.2 | 68550.5 | ||||||||||||||||||
1.4 | −41.4499 | −168.991 | 1206.10 | −173.857 | 7004.67 | 1282.71 | −28770.2 | 8875.03 | 7206.38 | ||||||||||||||||||
1.5 | −41.2529 | 209.636 | 1189.80 | −331.195 | −8648.07 | 2562.12 | −27961.2 | 24264.1 | 13662.8 | ||||||||||||||||||
1.6 | −40.7005 | −229.764 | 1144.53 | −2001.03 | 9351.50 | −7882.09 | −25744.4 | 33108.3 | 81443.0 | ||||||||||||||||||
1.7 | −39.1065 | −17.2889 | 1017.32 | 2451.89 | 676.107 | 9796.00 | −19761.4 | −19384.1 | −95884.8 | ||||||||||||||||||
1.8 | −38.7497 | 22.9806 | 989.541 | −1153.36 | −890.491 | −1095.21 | −18504.6 | −19154.9 | 44692.5 | ||||||||||||||||||
1.9 | −37.7791 | 61.3514 | 915.258 | −812.282 | −2317.80 | 9612.97 | −15234.7 | −15919.0 | 30687.2 | ||||||||||||||||||
1.10 | −37.7547 | 255.082 | 913.417 | 2014.92 | −9630.55 | −1677.30 | −15155.4 | 45383.9 | −76072.6 | ||||||||||||||||||
1.11 | −37.6864 | −151.403 | 908.263 | 1470.80 | 5705.84 | 10788.4 | −14933.7 | 3239.97 | −55429.1 | ||||||||||||||||||
1.12 | −37.1794 | −114.877 | 870.305 | 583.545 | 4271.07 | 4499.03 | −13321.6 | −6486.18 | −21695.8 | ||||||||||||||||||
1.13 | −35.7948 | 125.776 | 769.270 | −327.580 | −4502.12 | 1820.37 | −9208.93 | −3863.47 | 11725.7 | ||||||||||||||||||
1.14 | −32.1640 | −36.2624 | 522.526 | −2127.99 | 1166.35 | −6745.62 | −338.551 | −18368.0 | 68444.9 | ||||||||||||||||||
1.15 | −31.7734 | −146.785 | 497.551 | 1486.83 | 4663.85 | −4360.64 | 459.090 | 1862.72 | −47241.8 | ||||||||||||||||||
1.16 | −30.1558 | −183.098 | 397.372 | −2217.56 | 5521.45 | −821.704 | 3456.69 | 13841.7 | 66872.4 | ||||||||||||||||||
1.17 | −29.6607 | −146.482 | 367.760 | 2338.75 | 4344.77 | −2788.08 | 4278.27 | 1773.98 | −69369.1 | ||||||||||||||||||
1.18 | −29.1507 | 135.025 | 337.761 | 819.543 | −3936.06 | −12241.7 | 5079.18 | −1451.32 | −23890.2 | ||||||||||||||||||
1.19 | −28.5483 | −27.9042 | 303.004 | 1083.20 | 796.615 | −5365.56 | 5966.48 | −18904.4 | −30923.6 | ||||||||||||||||||
1.20 | −25.8744 | 201.470 | 157.486 | 2132.18 | −5212.93 | 11367.3 | 9172.85 | 20907.2 | −55169.0 | ||||||||||||||||||
See all 82 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(211\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 211.10.a.b | ✓ | 82 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
211.10.a.b | ✓ | 82 | 1.a | even | 1 | 1 | trivial |