Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [177,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("177.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 177 = 3 \cdot 59 \) |
Weight: | \( k \) | \(=\) | \( 14 \) |
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: | \(189.798744245\) |
Analytic rank: | \(1\) |
Dimension: | \(31\) |
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 | −169.906 | −729.000 | 20675.9 | 41498.5 | 123861. | 222317. | −2.12109e6 | 531441. | −7.05083e6 | ||||||||||||||||||
1.2 | −169.081 | −729.000 | 20396.4 | 15031.4 | 123260. | −257399. | −2.06354e6 | 531441. | −2.54153e6 | ||||||||||||||||||
1.3 | −159.673 | −729.000 | 17303.4 | −10788.6 | 116402. | 188818. | −1.45485e6 | 531441. | 1.72264e6 | ||||||||||||||||||
1.4 | −151.259 | −729.000 | 14687.4 | 64912.3 | 110268. | −35238.1 | −982489. | 531441. | −9.81859e6 | ||||||||||||||||||
1.5 | −149.360 | −729.000 | 14116.3 | −48711.9 | 108883. | −300158. | −884852. | 531441. | 7.27559e6 | ||||||||||||||||||
1.6 | −135.114 | −729.000 | 10063.7 | −36751.1 | 98497.9 | −177254. | −252896. | 531441. | 4.96558e6 | ||||||||||||||||||
1.7 | −117.761 | −729.000 | 5675.60 | −28297.1 | 85847.6 | 297580. | 296333. | 531441. | 3.33229e6 | ||||||||||||||||||
1.8 | −98.0136 | −729.000 | 1414.67 | 43954.8 | 71451.9 | −396678. | 664271. | 531441. | −4.30817e6 | ||||||||||||||||||
1.9 | −80.4989 | −729.000 | −1711.93 | −19074.6 | 58683.7 | −213612. | 797255. | 531441. | 1.53548e6 | ||||||||||||||||||
1.10 | −73.1261 | −729.000 | −2844.57 | −37898.1 | 53309.0 | 370148. | 807062. | 531441. | 2.77134e6 | ||||||||||||||||||
1.11 | −66.2972 | −729.000 | −3796.68 | 24188.3 | 48330.7 | −395052. | 794816. | 531441. | −1.60362e6 | ||||||||||||||||||
1.12 | −63.8588 | −729.000 | −4114.05 | 18414.6 | 46553.1 | 120263. | 785850. | 531441. | −1.17593e6 | ||||||||||||||||||
1.13 | −57.5851 | −729.000 | −4875.95 | 31633.2 | 41979.6 | 267144. | 752520. | 531441. | −1.82160e6 | ||||||||||||||||||
1.14 | −42.0742 | −729.000 | −6421.76 | −29646.8 | 30672.1 | 519485. | 614862. | 531441. | 1.24737e6 | ||||||||||||||||||
1.15 | −32.6149 | −729.000 | −7128.27 | −49381.7 | 23776.2 | −593230. | 499669. | 531441. | 1.61058e6 | ||||||||||||||||||
1.16 | 13.8965 | −729.000 | −7998.89 | 24353.7 | −10130.5 | −38818.7 | −224997. | 531441. | 338431. | ||||||||||||||||||
1.17 | 20.8810 | −729.000 | −7755.98 | 27865.9 | −15222.2 | 61118.3 | −333010. | 531441. | 581867. | ||||||||||||||||||
1.18 | 32.0352 | −729.000 | −7165.75 | 52348.2 | −23353.7 | 485703. | −491988. | 531441. | 1.67699e6 | ||||||||||||||||||
1.19 | 39.4833 | −729.000 | −6633.07 | 39747.6 | −28783.4 | −539057. | −585343. | 531441. | 1.56937e6 | ||||||||||||||||||
1.20 | 52.9360 | −729.000 | −5389.78 | −33150.0 | −38590.4 | −413060. | −718965. | 531441. | −1.75483e6 | ||||||||||||||||||
See all 31 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 | 177.14.a.b | ✓ | 31 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
177.14.a.b | ✓ | 31 | 1.a | even | 1 | 1 | trivial |