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: | \(1\) |
Dimension: | \(26\) |
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 | −87.0020 | −243.000 | 5521.36 | −497.667 | 21141.5 | 10880.7 | −302189. | 59049.0 | 43298.0 | ||||||||||||||||||
1.2 | −86.0142 | −243.000 | 5350.44 | 12869.1 | 20901.4 | −72810.3 | −284057. | 59049.0 | −1.10693e6 | ||||||||||||||||||
1.3 | −70.0902 | −243.000 | 2864.64 | 3425.74 | 17031.9 | −19416.7 | −57238.6 | 59049.0 | −240111. | ||||||||||||||||||
1.4 | −68.1728 | −243.000 | 2599.53 | −12352.0 | 16566.0 | 47034.4 | −37599.4 | 59049.0 | 842068. | ||||||||||||||||||
1.5 | −66.9000 | −243.000 | 2427.61 | −5686.53 | 16256.7 | 26687.5 | −25395.7 | 59049.0 | 380429. | ||||||||||||||||||
1.6 | −60.8905 | −243.000 | 1659.66 | −4960.99 | 14796.4 | −87058.0 | 23646.5 | 59049.0 | 302077. | ||||||||||||||||||
1.7 | −49.4794 | −243.000 | 400.209 | 6608.87 | 12023.5 | −17933.1 | 81531.7 | 59049.0 | −327003. | ||||||||||||||||||
1.8 | −43.6565 | −243.000 | −142.110 | 7124.43 | 10608.5 | 50312.1 | 95612.5 | 59049.0 | −311028. | ||||||||||||||||||
1.9 | −42.1259 | −243.000 | −273.404 | −6338.85 | 10236.6 | −22948.3 | 97791.4 | 59049.0 | 267030. | ||||||||||||||||||
1.10 | −35.6107 | −243.000 | −779.875 | −1849.70 | 8653.41 | 64152.0 | 100703. | 59049.0 | 65869.2 | ||||||||||||||||||
1.11 | −26.3367 | −243.000 | −1354.38 | 12487.3 | 6399.81 | −48376.8 | 89607.3 | 59049.0 | −328874. | ||||||||||||||||||
1.12 | −16.5456 | −243.000 | −1774.24 | 2844.36 | 4020.58 | 41460.7 | 63241.3 | 59049.0 | −47061.6 | ||||||||||||||||||
1.13 | −10.8994 | −243.000 | −1929.20 | 3570.42 | 2648.56 | −66491.5 | 43349.2 | 59049.0 | −38915.5 | ||||||||||||||||||
1.14 | 9.61747 | −243.000 | −1955.50 | −8978.47 | −2337.04 | 1935.43 | −38503.6 | 59049.0 | −86350.1 | ||||||||||||||||||
1.15 | 10.8639 | −243.000 | −1929.97 | 8237.50 | −2639.94 | 85962.3 | −43216.5 | 59049.0 | 89491.8 | ||||||||||||||||||
1.16 | 16.4601 | −243.000 | −1777.06 | −11170.6 | −3999.81 | −64730.0 | −62961.0 | 59049.0 | −183870. | ||||||||||||||||||
1.17 | 20.9819 | −243.000 | −1607.76 | 5799.05 | −5098.61 | 22173.1 | −76704.8 | 59049.0 | 121675. | ||||||||||||||||||
1.18 | 27.2209 | −243.000 | −1307.02 | −8750.00 | −6614.68 | −68225.0 | −91326.7 | 59049.0 | −238183. | ||||||||||||||||||
1.19 | 38.1800 | −243.000 | −590.285 | 118.373 | −9277.75 | −24998.6 | −100730. | 59049.0 | 4519.50 | ||||||||||||||||||
1.20 | 40.1266 | −243.000 | −437.857 | 10858.4 | −9750.76 | −2158.10 | −99749.0 | 59049.0 | 435712. | ||||||||||||||||||
See all 26 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.a | ✓ | 26 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.12.a.a | ✓ | 26 | 1.a | even | 1 | 1 | trivial |