Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,6,Mod(440,441)]
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([1, 1]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.440");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.c (of order \(2\), degree \(1\), 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: | \(70.7292645375\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 63) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 7\nu^{2} - 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{3} + 28\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 7\beta_1 ) / 14 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 7 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(344\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 440.1 |
|
− | 2.82843i | 0 | 24.0000 | −51.4393 | 0 | 0 | − | 158.392i | 0 | 145.492i | ||||||||||||||||||||||||||||
| 440.2 | − | 2.82843i | 0 | 24.0000 | 51.4393 | 0 | 0 | − | 158.392i | 0 | − | 145.492i | ||||||||||||||||||||||||||||
| 440.3 | 2.82843i | 0 | 24.0000 | −51.4393 | 0 | 0 | 158.392i | 0 | − | 145.492i | ||||||||||||||||||||||||||||||
| 440.4 | 2.82843i | 0 | 24.0000 | 51.4393 | 0 | 0 | 158.392i | 0 | 145.492i | |||||||||||||||||||||||||||||||
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.c.a | 4 | |
| 3.b | odd | 2 | 1 | inner | 441.6.c.a | 4 | |
| 7.b | odd | 2 | 1 | inner | 441.6.c.a | 4 | |
| 7.c | even | 3 | 1 | 63.6.p.a | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 63.6.p.a | ✓ | 4 | |
| 21.c | even | 2 | 1 | inner | 441.6.c.a | 4 | |
| 21.g | even | 6 | 1 | 63.6.p.a | ✓ | 4 | |
| 21.h | odd | 6 | 1 | 63.6.p.a | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.6.p.a | ✓ | 4 | 7.c | even | 3 | 1 | |
| 63.6.p.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 63.6.p.a | ✓ | 4 | 21.g | even | 6 | 1 | |
| 63.6.p.a | ✓ | 4 | 21.h | odd | 6 | 1 | |
| 441.6.c.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 441.6.c.a | 4 | 3.b | odd | 2 | 1 | inner | |
| 441.6.c.a | 4 | 7.b | odd | 2 | 1 | inner | |
| 441.6.c.a | 4 | 21.c | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 8 \)
acting on \(S_{6}^{\mathrm{new}}(441, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 2646)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 334562)^{2} \)
|
| $13$ |
\( (T^{2} + 1326675)^{2} \)
|
| $17$ |
\( (T^{2} - 142296)^{2} \)
|
| $19$ |
\( (T^{2} + 3006003)^{2} \)
|
| $23$ |
\( (T^{2} + 21255200)^{2} \)
|
| $29$ |
\( (T^{2} + 21177032)^{2} \)
|
| $31$ |
\( (T^{2} + 5140443)^{2} \)
|
| $37$ |
\( (T - 203)^{4} \)
|
| $41$ |
\( (T^{2} - 7350)^{2} \)
|
| $43$ |
\( (T - 4697)^{4} \)
|
| $47$ |
\( (T^{2} - 575417094)^{2} \)
|
| $53$ |
\( (T^{2} + 55041032)^{2} \)
|
| $59$ |
\( (T^{2} - 24725400)^{2} \)
|
| $61$ |
\( (T^{2} + 303973068)^{2} \)
|
| $67$ |
\( (T - 30197)^{4} \)
|
| $71$ |
\( (T^{2} + 15691202)^{2} \)
|
| $73$ |
\( (T^{2} + 2844626547)^{2} \)
|
| $79$ |
\( (T + 39385)^{4} \)
|
| $83$ |
\( (T^{2} - 4040103606)^{2} \)
|
| $89$ |
\( (T^{2} - 11105354904)^{2} \)
|
| $97$ |
\( (T^{2} + 4248300)^{2} \)
|
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