Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(137.761796238\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 5825x^{2} + 5826x + 8447739 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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
\( x^{4} - 2x^{3} - 5825x^{2} + 5826x + 8447739 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -8\nu^{3} + 12\nu^{2} + 23352\nu - 11678 ) / 11627 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -12\nu^{3} + 18\nu^{2} + 104790\nu - 52398 ) / 11627 \)
|
| \(\beta_{3}\) | \(=\) |
\( 12\nu^{2} - 12\nu - 34956 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{2} - 3\beta _1 + 6 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - 3\beta _1 + 34962 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 3894\beta_{2} - 17468\beta _1 + 34960 ) / 8 \)
|
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 |
|
−7.21110 | 0 | −76.0000 | −323.666 | 0 | 0 | 1471.06 | 0 | 2333.99 | ||||||||||||||||||||||||||||||
| 1.2 | −7.21110 | 0 | −76.0000 | 323.666 | 0 | 0 | 1471.06 | 0 | −2333.99 | |||||||||||||||||||||||||||||||
| 1.3 | 7.21110 | 0 | −76.0000 | −323.666 | 0 | 0 | −1471.06 | 0 | −2333.99 | |||||||||||||||||||||||||||||||
| 1.4 | 7.21110 | 0 | −76.0000 | 323.666 | 0 | 0 | −1471.06 | 0 | 2333.99 | |||||||||||||||||||||||||||||||
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 | 441.8.a.o | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 441.8.a.o | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 441.8.a.o | ✓ | 4 |
| 21.c | even | 2 | 1 | inner | 441.8.a.o | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 441.8.a.o | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 441.8.a.o | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 441.8.a.o | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 441.8.a.o | ✓ | 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{2} - 52 \)
|
|
\( T_{5}^{2} - 104760 \)
|
|
\( T_{13}^{2} - 136188000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 52)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 104760)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 2185300)^{2} \)
|
| $13$ |
\( (T^{2} - 136188000)^{2} \)
|
| $17$ |
\( (T^{2} - 17704440)^{2} \)
|
| $19$ |
\( (T^{2} - 1574333280)^{2} \)
|
| $23$ |
\( (T^{2} - 6158863828)^{2} \)
|
| $29$ |
\( (T^{2} - 10707970000)^{2} \)
|
| $31$ |
\( (T^{2} - 15302083680)^{2} \)
|
| $37$ |
\( (T + 115050)^{4} \)
|
| $41$ |
\( (T^{2} - 689239611000)^{2} \)
|
| $43$ |
\( (T - 377940)^{4} \)
|
| $47$ |
\( (T^{2} - 915386175360)^{2} \)
|
| $53$ |
\( (T^{2} - 147737366608)^{2} \)
|
| $59$ |
\( (T^{2} - 155924784000)^{2} \)
|
| $61$ |
\( (T^{2} - 5656492314720)^{2} \)
|
| $67$ |
\( (T - 1129180)^{4} \)
|
| $71$ |
\( (T^{2} - 8627086479700)^{2} \)
|
| $73$ |
\( (T^{2} - 1708342272000)^{2} \)
|
| $79$ |
\( (T + 2418184)^{4} \)
|
| $83$ |
\( (T^{2} - 16557861075840)^{2} \)
|
| $89$ |
\( (T^{2} - 63680359059000)^{2} \)
|
| $97$ |
\( (T^{2} - 32698738800000)^{2} \)
|
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