Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,10,Mod(1,177)]
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([0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 177.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: | \(91.1613430010\) |
Analytic rank: | \(1\) |
Dimension: | \(21\) |
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 | −43.9134 | 81.0000 | 1416.39 | −993.635 | −3556.99 | 3425.90 | −39714.8 | 6561.00 | 43633.9 | ||||||||||||||||||
1.2 | −43.5628 | 81.0000 | 1385.72 | 1597.21 | −3528.58 | −10966.0 | −38061.5 | 6561.00 | −69578.8 | ||||||||||||||||||
1.3 | −35.8479 | 81.0000 | 773.074 | 1881.24 | −2903.68 | −9737.47 | −9358.98 | 6561.00 | −67438.4 | ||||||||||||||||||
1.4 | −31.8892 | 81.0000 | 504.921 | −1889.12 | −2583.03 | 12272.3 | 225.735 | 6561.00 | 60242.4 | ||||||||||||||||||
1.5 | −29.8118 | 81.0000 | 376.741 | 1198.15 | −2414.75 | 3753.28 | 4032.31 | 6561.00 | −35719.0 | ||||||||||||||||||
1.6 | −26.8579 | 81.0000 | 209.348 | −2742.53 | −2175.49 | −8878.91 | 8128.61 | 6561.00 | 73658.6 | ||||||||||||||||||
1.7 | −24.8760 | 81.0000 | 106.816 | −222.031 | −2014.96 | −2494.49 | 10079.4 | 6561.00 | 5523.25 | ||||||||||||||||||
1.8 | −18.1702 | 81.0000 | −181.842 | 1096.53 | −1471.79 | −2885.78 | 12607.3 | 6561.00 | −19924.2 | ||||||||||||||||||
1.9 | −16.6893 | 81.0000 | −233.469 | −1238.34 | −1351.83 | 5162.18 | 12441.3 | 6561.00 | 20667.0 | ||||||||||||||||||
1.10 | −12.7580 | 81.0000 | −349.234 | 1739.29 | −1033.40 | 7509.10 | 10987.6 | 6561.00 | −22189.8 | ||||||||||||||||||
1.11 | −3.19317 | 81.0000 | −501.804 | −2710.01 | −258.647 | −1841.65 | 3237.25 | 6561.00 | 8653.54 | ||||||||||||||||||
1.12 | 1.56630 | 81.0000 | −509.547 | −1262.61 | 126.871 | −7655.10 | −1600.05 | 6561.00 | −1977.63 | ||||||||||||||||||
1.13 | 2.31191 | 81.0000 | −506.655 | −66.4563 | 187.265 | −1904.69 | −2355.04 | 6561.00 | −153.641 | ||||||||||||||||||
1.14 | 6.97380 | 81.0000 | −463.366 | 1656.32 | 564.878 | 2037.84 | −6802.01 | 6561.00 | 11550.8 | ||||||||||||||||||
1.15 | 15.9818 | 81.0000 | −256.580 | 61.0966 | 1294.53 | 4757.03 | −12283.3 | 6561.00 | 976.437 | ||||||||||||||||||
1.16 | 24.3661 | 81.0000 | 81.7051 | 1969.67 | 1973.65 | −11418.4 | −10484.6 | 6561.00 | 47993.2 | ||||||||||||||||||
1.17 | 27.0996 | 81.0000 | 222.388 | −1517.97 | 2195.07 | 2210.22 | −7848.37 | 6561.00 | −41136.3 | ||||||||||||||||||
1.18 | 30.4984 | 81.0000 | 418.150 | 1050.93 | 2470.37 | −5890.68 | −2862.26 | 6561.00 | 32051.7 | ||||||||||||||||||
1.19 | 35.2875 | 81.0000 | 733.210 | 388.913 | 2858.29 | −2376.48 | 7805.95 | 6561.00 | 13723.8 | ||||||||||||||||||
1.20 | 37.7088 | 81.0000 | 909.951 | −1229.57 | 3054.41 | −1968.46 | 15006.2 | 6561.00 | −46365.5 | ||||||||||||||||||
See all 21 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(59\) | \(-1\) |
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 | 177.10.a.a | ✓ | 21 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.10.a.a | ✓ | 21 | 1.a | even | 1 | 1 | trivial |