Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.37");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(64.8945153566\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 27343x^{2} + 27342x + 747584964 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 7^{2} \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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
\( x^{4} - x^{3} + 27343x^{2} + 27342x + 747584964 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 27343\nu^{2} - 27343\nu + 747584964 ) / 747612306 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 54686\nu^{2} + 2616697757\nu - 1495169928 ) / 373806153 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{3} + 300766 ) / 27343 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta_1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 191398\beta _1 - 191398 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 27343\beta_{3} - 300766 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) |
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 |
|
−8.00000 | + | 13.8564i | 0 | −128.000 | − | 221.703i | −818.992 | + | 1418.54i | 0 | −5958.47 | − | 2202.33i | 4096.00 | 0 | −13103.9 | − | 22696.6i | ||||||||||||||||||||
| 37.2 | −8.00000 | + | 13.8564i | 0 | −128.000 | − | 221.703i | 338.492 | − | 586.286i | 0 | −1328.53 | − | 6211.97i | 4096.00 | 0 | 5415.88 | + | 9380.57i | |||||||||||||||||||||
| 109.1 | −8.00000 | − | 13.8564i | 0 | −128.000 | + | 221.703i | −818.992 | − | 1418.54i | 0 | −5958.47 | + | 2202.33i | 4096.00 | 0 | −13103.9 | + | 22696.6i | |||||||||||||||||||||
| 109.2 | −8.00000 | − | 13.8564i | 0 | −128.000 | + | 221.703i | 338.492 | + | 586.286i | 0 | −1328.53 | + | 6211.97i | 4096.00 | 0 | 5415.88 | − | 9380.57i | |||||||||||||||||||||
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 | 126.10.g.b | 4 | |
| 3.b | odd | 2 | 1 | 42.10.e.c | ✓ | 4 | |
| 7.c | even | 3 | 1 | inner | 126.10.g.b | 4 | |
| 21.g | even | 6 | 1 | 294.10.a.j | 2 | ||
| 21.h | odd | 6 | 1 | 42.10.e.c | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 294.10.a.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.10.e.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 42.10.e.c | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 126.10.g.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 126.10.g.b | 4 | 7.c | even | 3 | 1 | inner | |
| 294.10.a.j | 2 | 21.g | even | 6 | 1 | ||
| 294.10.a.k | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 961T_{5}^{3} + 2032411T_{5}^{2} - 1065643290T_{5} + 1229637032100 \)
acting on \(S_{10}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 256)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 1229637032100 \)
|
| $7$ |
\( T^{4} + \cdots + 16\!\cdots\!49 \)
|
| $11$ |
\( T^{4} + \cdots + 60\!\cdots\!76 \)
|
| $13$ |
\( (T^{2} + 22639 T - 25757569920)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 77\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots + 72\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 19\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} - 424441 T - 628439508420)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 69\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots + 91\!\cdots\!24 \)
|
| $41$ |
\( (T^{2} + \cdots + 143646184498320)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots - 538220637195050)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 40\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 29\!\cdots\!76 \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!96 \)
|
| $61$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
|
| $71$ |
\( (T^{2} + \cdots - 53\!\cdots\!24)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 31\!\cdots\!16 \)
|
| $79$ |
\( T^{4} + \cdots + 13\!\cdots\!09 \)
|
| $83$ |
\( (T^{2} + \cdots - 52\!\cdots\!30)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 23\!\cdots\!16 \)
|
| $97$ |
\( (T^{2} + \cdots + 15\!\cdots\!22)^{2} \)
|
| show more | |
| show less |