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: | \(27\) |
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 | −83.3973 | −243.000 | 4907.11 | −4959.60 | 20265.5 | 14303.6 | −238442. | 59049.0 | 413617. | ||||||||||||||||||
1.2 | −81.8884 | −243.000 | 4657.72 | −12268.5 | 19898.9 | −65634.7 | −213706. | 59049.0 | 1.00465e6 | ||||||||||||||||||
1.3 | −79.2286 | −243.000 | 4229.17 | 9672.04 | 19252.5 | 84867.6 | −172811. | 59049.0 | −766302. | ||||||||||||||||||
1.4 | −70.9666 | −243.000 | 2988.26 | 3032.24 | 17244.9 | −62069.7 | −66727.2 | 59049.0 | −215188. | ||||||||||||||||||
1.5 | −66.1318 | −243.000 | 2325.41 | 6882.52 | 16070.0 | 7668.92 | −18345.7 | 59049.0 | −455153. | ||||||||||||||||||
1.6 | −64.1897 | −243.000 | 2072.32 | −6307.39 | 15598.1 | 84626.9 | −1560.87 | 59049.0 | 404869. | ||||||||||||||||||
1.7 | −58.2273 | −243.000 | 1342.41 | 11765.2 | 14149.2 | 24587.2 | 41084.4 | 59049.0 | −685053. | ||||||||||||||||||
1.8 | −41.9561 | −243.000 | −287.686 | −4799.69 | 10195.3 | −40967.8 | 97996.3 | 59049.0 | 201376. | ||||||||||||||||||
1.9 | −41.5867 | −243.000 | −318.543 | 9657.75 | 10105.6 | −11578.9 | 98416.8 | 59049.0 | −401634. | ||||||||||||||||||
1.10 | −38.5064 | −243.000 | −565.255 | −13229.8 | 9357.06 | −8923.95 | 100627. | 59049.0 | 509431. | ||||||||||||||||||
1.11 | −11.8207 | −243.000 | −1908.27 | −8832.11 | 2872.42 | 19217.9 | 46765.7 | 59049.0 | 104401. | ||||||||||||||||||
1.12 | −10.6257 | −243.000 | −1935.10 | −9082.46 | 2582.03 | 57618.4 | 42323.0 | 59049.0 | 96507.1 | ||||||||||||||||||
1.13 | −8.53597 | −243.000 | −1975.14 | −453.475 | 2074.24 | −60294.7 | 34341.4 | 59049.0 | 3870.85 | ||||||||||||||||||
1.14 | −5.14065 | −243.000 | −2021.57 | 3789.43 | 1249.18 | −36302.1 | 20920.3 | 59049.0 | −19480.2 | ||||||||||||||||||
1.15 | −3.32183 | −243.000 | −2036.97 | 10388.6 | 807.204 | 36007.4 | 13569.5 | 59049.0 | −34509.2 | ||||||||||||||||||
1.16 | 7.89171 | −243.000 | −1985.72 | −1405.99 | −1917.69 | 25105.4 | −31832.9 | 59049.0 | −11095.6 | ||||||||||||||||||
1.17 | 18.0234 | −243.000 | −1723.16 | −3879.65 | −4379.69 | 32120.2 | −67969.1 | 59049.0 | −69924.5 | ||||||||||||||||||
1.18 | 24.9877 | −243.000 | −1423.61 | 11346.7 | −6072.02 | −46106.7 | −86747.8 | 59049.0 | 283528. | ||||||||||||||||||
1.19 | 41.1468 | −243.000 | −354.944 | −4337.67 | −9998.66 | 64289.5 | −98873.4 | 59049.0 | −178481. | ||||||||||||||||||
1.20 | 43.4861 | −243.000 | −156.959 | 1833.80 | −10567.1 | 33856.7 | −95885.1 | 59049.0 | 79745.0 | ||||||||||||||||||
See all 27 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.c | ✓ | 27 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.12.a.c | ✓ | 27 | 1.a | even | 1 | 1 | trivial |