Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 12 \) |
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: | \(135.996742959\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
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 | −88.1033 | 243.000 | 5714.20 | 1093.28 | −21409.1 | −20521.5 | −323004. | 59049.0 | −96322.0 | ||||||||||||||||||
1.2 | −79.8853 | 243.000 | 4333.66 | −5377.20 | −19412.1 | −58607.4 | −182591. | 59049.0 | 429560. | ||||||||||||||||||
1.3 | −76.9358 | 243.000 | 3871.11 | 1565.71 | −18695.4 | 51126.9 | −140263. | 59049.0 | −120459. | ||||||||||||||||||
1.4 | −67.4662 | 243.000 | 2503.69 | −169.543 | −16394.3 | 65772.9 | −30743.5 | 59049.0 | 11438.4 | ||||||||||||||||||
1.5 | −67.3482 | 243.000 | 2487.78 | 9961.02 | −16365.6 | −37608.5 | −29618.6 | 59049.0 | −670857. | ||||||||||||||||||
1.6 | −47.6933 | 243.000 | 226.652 | 10067.2 | −11589.5 | 19897.8 | 86866.1 | 59049.0 | −480139. | ||||||||||||||||||
1.7 | −44.5190 | 243.000 | −66.0554 | −2490.42 | −10818.1 | −10217.0 | 94115.7 | 59049.0 | 110871. | ||||||||||||||||||
1.8 | −40.3996 | 243.000 | −415.876 | 13699.0 | −9817.09 | 81088.1 | 99539.5 | 59049.0 | −553432. | ||||||||||||||||||
1.9 | −39.1439 | 243.000 | −515.752 | 158.720 | −9511.98 | −70159.1 | 100355. | 59049.0 | −6212.92 | ||||||||||||||||||
1.10 | −33.4729 | 243.000 | −927.563 | −11827.5 | −8133.92 | 55304.7 | 99600.8 | 59049.0 | 395902. | ||||||||||||||||||
1.11 | −33.1334 | 243.000 | −950.178 | −7615.06 | −8051.42 | 31854.1 | 99339.8 | 59049.0 | 252313. | ||||||||||||||||||
1.12 | −12.4112 | 243.000 | −1893.96 | −2629.42 | −3015.92 | 23296.0 | 48924.4 | 59049.0 | 32634.3 | ||||||||||||||||||
1.13 | −7.28530 | 243.000 | −1994.92 | 6136.55 | −1770.33 | −33718.8 | 29453.9 | 59049.0 | −44706.7 | ||||||||||||||||||
1.14 | −5.21206 | 243.000 | −2020.83 | −8551.43 | −1266.53 | −16524.4 | 21207.0 | 59049.0 | 44570.6 | ||||||||||||||||||
1.15 | 10.5323 | 243.000 | −1937.07 | 8300.31 | 2559.34 | 46494.5 | −41971.8 | 59049.0 | 87421.0 | ||||||||||||||||||
1.16 | 23.0914 | 243.000 | −1514.79 | 32.6758 | 5611.21 | −80813.9 | −82269.7 | 59049.0 | 754.528 | ||||||||||||||||||
1.17 | 23.7768 | 243.000 | −1482.66 | 6921.85 | 5777.76 | 24486.4 | −83947.8 | 59049.0 | 164579. | ||||||||||||||||||
1.18 | 31.6204 | 243.000 | −1048.15 | −7920.33 | 7683.75 | 55386.7 | −97901.5 | 59049.0 | −250444. | ||||||||||||||||||
1.19 | 31.7299 | 243.000 | −1041.21 | 1823.69 | 7710.37 | −52038.7 | −98020.5 | 59049.0 | 57865.4 | ||||||||||||||||||
1.20 | 45.0643 | 243.000 | −17.2112 | −12444.4 | 10950.6 | −11322.2 | −93067.2 | 59049.0 | −560797. | ||||||||||||||||||
See all 28 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 | 177.12.a.d | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.12.a.d | ✓ | 28 | 1.a | even | 1 | 1 | trivial |