Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,8,Mod(37,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.g (of order \(3\), 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: | \(39.3605132110\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5497})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 1375x^{2} + 1374x + 1887876 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 1375x^{2} + 1374x + 1887876 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 1375\nu^{2} - 1375\nu + 1887876 ) / 1889250 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 2749 ) / 1375 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 1374\beta_{2} + \beta _1 - 1375 \)
|
| \(\nu^{3}\) | \(=\) |
\( 1375\beta_{3} - 2749 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
−4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −206.427 | + | 357.542i | 0 | −130.492 | + | 898.062i | 512.000 | 0 | −1651.42 | − | 2860.34i | ||||||||||||||||||||
| 37.2 | −4.00000 | + | 6.92820i | 0 | −32.0000 | − | 55.4256i | −21.0728 | + | 36.4992i | 0 | 907.492 | − | 0.859351i | 512.000 | 0 | −168.582 | − | 291.993i | |||||||||||||||||||||
| 109.1 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −206.427 | − | 357.542i | 0 | −130.492 | − | 898.062i | 512.000 | 0 | −1651.42 | + | 2860.34i | |||||||||||||||||||||
| 109.2 | −4.00000 | − | 6.92820i | 0 | −32.0000 | + | 55.4256i | −21.0728 | − | 36.4992i | 0 | 907.492 | + | 0.859351i | 512.000 | 0 | −168.582 | + | 291.993i | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 126.8.g.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | 126.8.g.g | yes | 4 | |
| 7.c | even | 3 | 1 | inner | 126.8.g.c | ✓ | 4 |
| 21.h | odd | 6 | 1 | 126.8.g.g | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.8.g.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 126.8.g.c | ✓ | 4 | 7.c | even | 3 | 1 | inner |
| 126.8.g.g | yes | 4 | 3.b | odd | 2 | 1 | |
| 126.8.g.g | yes | 4 | 21.h | odd | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 455T_{5}^{3} + 189625T_{5}^{2} + 7917000T_{5} + 302760000 \)
acting on \(S_{8}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 455 T^{3} + \cdots + 302760000 \)
|
| $7$ |
\( T^{4} + \cdots + 678223072849 \)
|
| $11$ |
\( T^{4} + \cdots + 121771401560064 \)
|
| $13$ |
\( (T^{2} + 20461 T + 83190474)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 88\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots + 29\!\cdots\!76 \)
|
| $23$ |
\( T^{4} + \cdots + 16\!\cdots\!96 \)
|
| $29$ |
\( (T^{2} - 164357 T - 16134224592)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 15\!\cdots\!09 \)
|
| $37$ |
\( T^{4} + \cdots + 27\!\cdots\!84 \)
|
| $41$ |
\( (T^{2} - 1231944 T + 331907635584)^{2} \)
|
| $43$ |
\( (T^{2} + 1205387 T + 290266272844)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 20\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 67\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 31\!\cdots\!36 \)
|
| $61$ |
\( T^{4} + \cdots + 27\!\cdots\!64 \)
|
| $67$ |
\( T^{4} + \cdots + 25\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} - 5779648 T - 412483025472)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 43\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots + 24\!\cdots\!41 \)
|
| $83$ |
\( (T^{2} + \cdots + 2226269335944)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 65\!\cdots\!24 \)
|
| $97$ |
\( (T^{2} + \cdots - 19353954158134)^{2} \)
|
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