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: | \(\Q(\sqrt{2}, \sqrt{113})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 7 \) |
| Twist minimal: | no (minimal twist has level 49) |
| 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
\( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 172\nu - 139 ) / 105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 51\nu^{2} - 86\nu - 1558 ) / 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} - 67\nu + 34 ) / 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 7\beta _1 + 7 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 31 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 86\beta_{3} + 21\beta_{2} + 245\beta _1 + 441 ) / 7 \)
|
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 |
|
−2.81507 | 0 | −24.0754 | −45.9910 | 0 | 0 | 157.856 | 0 | 129.468 | ||||||||||||||||||||||||||||||
| 1.2 | −2.81507 | 0 | −24.0754 | 45.9910 | 0 | 0 | 157.856 | 0 | −129.468 | |||||||||||||||||||||||||||||||
| 1.3 | 7.81507 | 0 | 29.0754 | −74.2753 | 0 | 0 | −22.8562 | 0 | −580.467 | |||||||||||||||||||||||||||||||
| 1.4 | 7.81507 | 0 | 29.0754 | 74.2753 | 0 | 0 | −22.8562 | 0 | 580.467 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 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.z | 4 | |
| 3.b | odd | 2 | 1 | 49.6.a.g | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 441.6.a.z | 4 | |
| 12.b | even | 2 | 1 | 784.6.a.bf | 4 | ||
| 21.c | even | 2 | 1 | 49.6.a.g | ✓ | 4 | |
| 21.g | even | 6 | 2 | 49.6.c.h | 8 | ||
| 21.h | odd | 6 | 2 | 49.6.c.h | 8 | ||
| 84.h | odd | 2 | 1 | 784.6.a.bf | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.6.a.g | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 49.6.a.g | ✓ | 4 | 21.c | even | 2 | 1 | |
| 49.6.c.h | 8 | 21.g | even | 6 | 2 | ||
| 49.6.c.h | 8 | 21.h | odd | 6 | 2 | ||
| 441.6.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 441.6.a.z | 4 | 7.b | odd | 2 | 1 | inner | |
| 784.6.a.bf | 4 | 12.b | even | 2 | 1 | ||
| 784.6.a.bf | 4 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{2} - 5T_{2} - 22 \)
|
|
\( T_{5}^{4} - 7632T_{5}^{2} + 11669056 \)
|
|
\( T_{13}^{4} - 1260672T_{13}^{2} + 76158337024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 5 T - 22)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 7632 T^{2} + 11669056 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 976 T + 234076)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 76158337024 \)
|
| $17$ |
\( T^{4} + \cdots + 1700202150724 \)
|
| $19$ |
\( T^{4} - 1359892 T^{2} + 56942116 \)
|
| $23$ |
\( (T^{2} - 3568 T - 160336)^{2} \)
|
| $29$ |
\( (T^{2} - 1676 T - 16661788)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 614334295349824 \)
|
| $37$ |
\( (T^{2} + 4604 T + 5234116)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 14\!\cdots\!56 \)
|
| $43$ |
\( (T^{2} - 10224 T - 998756)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
|
| $53$ |
\( (T^{2} - 51460 T + 472236292)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 11\!\cdots\!76 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!24 \)
|
| $67$ |
\( (T^{2} + 11448 T - 3789557552)^{2} \)
|
| $71$ |
\( (T^{2} - 76912 T + 953601968)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 20\!\cdots\!44 \)
|
| $79$ |
\( (T^{2} + 45344 T + 386672)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 13\!\cdots\!96 \)
|
| $97$ |
\( T^{4} + \cdots + 11\!\cdots\!44 \)
|
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