Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,3,Mod(46,171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(171, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("171.46");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 171 = 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 171.p (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(4.65941252056\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 8.0.19163381760000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{8} - 14x^{6} + 177x^{4} - 266x^{2} + 361 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\nu^{6} - 177\nu^{4} + 2478\nu^{2} - 361 ) / 3363 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -14\nu^{7} + 177\nu^{5} - 2478\nu^{3} + 3724\nu ) / 3363 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -20\nu^{6} + 413\nu^{4} - 4661\nu^{2} + 13167 ) / 3363 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 34\nu^{7} - 590\nu^{5} + 7139\nu^{3} - 23617\nu ) / 3363 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -39\nu^{6} + 413\nu^{4} - 4661\nu^{2} - 5320 ) / 3363 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -53\nu^{7} + 590\nu^{5} - 7139\nu^{3} - 11685\nu ) / 3363 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{6} + \beta_{4} + 7\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - \beta_{5} - 10\beta_{3} + 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{6} + 28\beta_{4} + 79\beta_{2} - 79 \)
|
| \(\nu^{5}\) | \(=\) |
\( 14\beta_{7} - 28\beta_{5} - 121\beta_{3} - 14\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -177\beta_{6} + 177\beta_{4} - 973 \)
|
| \(\nu^{7}\) | \(=\) |
\( -177\beta_{7} - 177\beta_{5} - 1858\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).
| \(n\) | \(20\) | \(154\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−3.05907 | − | 1.76616i | 0 | 4.23861 | + | 7.34149i | 0.533068 | − | 0.923301i | 0 | 0.477226 | − | 15.8150i | 0 | −3.26139 | + | 1.88296i | |||||||||||||||||||||||||||||||||
| 46.2 | −1.06868 | − | 0.617004i | 0 | −1.23861 | − | 2.14534i | 4.08850 | − | 7.08149i | 0 | −10.4772 | 7.99294i | 0 | −8.73861 | + | 5.04524i | |||||||||||||||||||||||||||||||||||
| 46.3 | 1.06868 | + | 0.617004i | 0 | −1.23861 | − | 2.14534i | −4.08850 | + | 7.08149i | 0 | −10.4772 | − | 7.99294i | 0 | −8.73861 | + | 5.04524i | ||||||||||||||||||||||||||||||||||
| 46.4 | 3.05907 | + | 1.76616i | 0 | 4.23861 | + | 7.34149i | −0.533068 | + | 0.923301i | 0 | 0.477226 | 15.8150i | 0 | −3.26139 | + | 1.88296i | |||||||||||||||||||||||||||||||||||
| 145.1 | −3.05907 | + | 1.76616i | 0 | 4.23861 | − | 7.34149i | 0.533068 | + | 0.923301i | 0 | 0.477226 | 15.8150i | 0 | −3.26139 | − | 1.88296i | |||||||||||||||||||||||||||||||||||
| 145.2 | −1.06868 | + | 0.617004i | 0 | −1.23861 | + | 2.14534i | 4.08850 | + | 7.08149i | 0 | −10.4772 | − | 7.99294i | 0 | −8.73861 | − | 5.04524i | ||||||||||||||||||||||||||||||||||
| 145.3 | 1.06868 | − | 0.617004i | 0 | −1.23861 | + | 2.14534i | −4.08850 | − | 7.08149i | 0 | −10.4772 | 7.99294i | 0 | −8.73861 | − | 5.04524i | |||||||||||||||||||||||||||||||||||
| 145.4 | 3.05907 | − | 1.76616i | 0 | 4.23861 | − | 7.34149i | −0.533068 | − | 0.923301i | 0 | 0.477226 | − | 15.8150i | 0 | −3.26139 | − | 1.88296i | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 19.d | odd | 6 | 1 | inner |
| 57.f | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 171.3.p.f | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 171.3.p.f | ✓ | 8 |
| 19.d | odd | 6 | 1 | inner | 171.3.p.f | ✓ | 8 |
| 57.f | even | 6 | 1 | inner | 171.3.p.f | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 171.3.p.f | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 171.3.p.f | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 171.3.p.f | ✓ | 8 | 19.d | odd | 6 | 1 | inner |
| 171.3.p.f | ✓ | 8 | 57.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(171, [\chi])\):
|
\( T_{2}^{8} - 14T_{2}^{6} + 177T_{2}^{4} - 266T_{2}^{2} + 361 \)
|
|
\( T_{5}^{8} + 68T_{5}^{6} + 4548T_{5}^{4} + 5168T_{5}^{2} + 5776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 14 T^{6} + \cdots + 361 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 68 T^{6} + \cdots + 5776 \)
|
| $7$ |
\( (T^{2} + 10 T - 5)^{4} \)
|
| $11$ |
\( (T^{4} - 140 T^{2} + 1900)^{2} \)
|
| $13$ |
\( (T^{4} + 30 T^{3} + \cdots + 1225)^{2} \)
|
| $17$ |
\( T^{8} + 168 T^{6} + \cdots + 7485696 \)
|
| $19$ |
\( (T^{4} + 48 T^{3} + \cdots + 130321)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 7687066773136 \)
|
| $29$ |
\( T^{8} + \cdots + 36100000000 \)
|
| $31$ |
\( (T^{4} + 746 T^{2} + 124609)^{2} \)
|
| $37$ |
\( (T^{4} + 4470 T^{2} + 2295225)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 14786560000 \)
|
| $43$ |
\( (T^{4} + 50 T^{3} + \cdots + 714025)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 14541836517376 \)
|
| $53$ |
\( T^{8} + \cdots + 32083206953616 \)
|
| $59$ |
\( T^{8} + \cdots + 33\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + 34 T^{3} + \cdots + 7349521)^{2} \)
|
| $67$ |
\( (T^{4} + 90 T^{3} + \cdots + 286225)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 390758009760000 \)
|
| $73$ |
\( (T^{4} + 30 T^{3} + \cdots + 90155025)^{2} \)
|
| $79$ |
\( (T^{4} - 210 T^{3} + \cdots + 4225)^{2} \)
|
| $83$ |
\( (T^{4} - 7208 T^{2} + 12527536)^{2} \)
|
| $89$ |
\( T^{8} + \cdots + 44\!\cdots\!00 \)
|
| $97$ |
\( (T^{4} + 420 T^{3} + \cdots + 197683600)^{2} \)
|
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