Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,6,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.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: | \(70.7292645375\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{19}, \sqrt{69})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 71x^{2} + 72x - 15 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 2x^{3} - 71x^{2} + 72x - 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 4\nu^{3} - 6\nu^{2} - 294\nu + 148 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 48\nu^{3} - 72\nu^{2} - 3360\nu + 1692 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( 24\nu^{2} - 24\nu - 864 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 12\beta _1 + 12 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 12\beta _1 + 876 ) / 24 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 50\beta_{2} - 572\beta _1 + 872 ) / 16 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−8.71780 | 0 | 44.0000 | −99.6795 | 0 | 0 | −104.614 | 0 | 868.986 | ||||||||||||||||||||||||||||||
| 1.2 | −8.71780 | 0 | 44.0000 | 99.6795 | 0 | 0 | −104.614 | 0 | −868.986 | |||||||||||||||||||||||||||||||
| 1.3 | 8.71780 | 0 | 44.0000 | −99.6795 | 0 | 0 | 104.614 | 0 | −868.986 | |||||||||||||||||||||||||||||||
| 1.4 | 8.71780 | 0 | 44.0000 | 99.6795 | 0 | 0 | 104.614 | 0 | 868.986 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 441.6.a.y | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 441.6.a.y | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 441.6.a.y | ✓ | 4 |
| 21.c | even | 2 | 1 | inner | 441.6.a.y | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 441.6.a.y | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 441.6.a.y | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 441.6.a.y | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 441.6.a.y | ✓ | 4 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{2} - 76 \)
|
|
\( T_{5}^{2} - 9936 \)
|
|
\( T_{13}^{2} - 755136 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 76)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 9936)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 140524)^{2} \)
|
| $13$ |
\( (T^{2} - 755136)^{2} \)
|
| $17$ |
\( (T^{2} - 1202256)^{2} \)
|
| $19$ |
\( (T^{2} - 755136)^{2} \)
|
| $23$ |
\( (T^{2} - 8837356)^{2} \)
|
| $29$ |
\( (T^{2} - 20392624)^{2} \)
|
| $31$ |
\( (T^{2} - 91371456)^{2} \)
|
| $37$ |
\( (T + 5466)^{4} \)
|
| $41$ |
\( (T^{2} - 97382736)^{2} \)
|
| $43$ |
\( (T - 12540)^{4} \)
|
| $47$ |
\( (T^{2} - 99360000)^{2} \)
|
| $53$ |
\( (T^{2} - 229568944)^{2} \)
|
| $59$ |
\( (T^{2} - 1820116224)^{2} \)
|
| $61$ |
\( (T^{2} - 2284286400)^{2} \)
|
| $67$ |
\( (T + 29996)^{4} \)
|
| $71$ |
\( (T^{2} - 3778461676)^{2} \)
|
| $73$ |
\( (T^{2} - 2368106496)^{2} \)
|
| $79$ |
\( (T - 80168)^{4} \)
|
| $83$ |
\( (T^{2} - 942568704)^{2} \)
|
| $89$ |
\( (T^{2} - 434014416)^{2} \)
|
| $97$ |
\( (T^{2} - 17908805376)^{2} \)
|
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