Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(244,441)]
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, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.244");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.d (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: | \(12.0163796583\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.2048.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 7 \) |
| Twist minimal: | no (minimal twist has level 49) |
| 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
\( x^{4} + 4x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} - 5\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -6\beta_{3} - 5\beta_1 ) / 7 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 244.1 |
|
−2.41421 | 0 | 1.82843 | − | 4.14386i | 0 | 0 | 5.24264 | 0 | 10.0042i | |||||||||||||||||||||||||||||
| 244.2 | −2.41421 | 0 | 1.82843 | 4.14386i | 0 | 0 | 5.24264 | 0 | − | 10.0042i | ||||||||||||||||||||||||||||||
| 244.3 | 0.414214 | 0 | −3.82843 | − | 4.77791i | 0 | 0 | −3.24264 | 0 | − | 1.97908i | |||||||||||||||||||||||||||||
| 244.4 | 0.414214 | 0 | −3.82843 | 4.77791i | 0 | 0 | −3.24264 | 0 | 1.97908i | |||||||||||||||||||||||||||||||
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 | 441.3.d.e | 4 | |
| 3.b | odd | 2 | 1 | 49.3.b.a | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 441.3.d.e | 4 | |
| 7.c | even | 3 | 2 | 441.3.m.l | 8 | ||
| 7.d | odd | 6 | 2 | 441.3.m.l | 8 | ||
| 12.b | even | 2 | 1 | 784.3.c.c | 4 | ||
| 21.c | even | 2 | 1 | 49.3.b.a | ✓ | 4 | |
| 21.g | even | 6 | 2 | 49.3.d.b | 8 | ||
| 21.h | odd | 6 | 2 | 49.3.d.b | 8 | ||
| 84.h | odd | 2 | 1 | 784.3.c.c | 4 | ||
| 84.j | odd | 6 | 2 | 784.3.s.h | 8 | ||
| 84.n | even | 6 | 2 | 784.3.s.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.3.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 49.3.b.a | ✓ | 4 | 21.c | even | 2 | 1 | |
| 49.3.d.b | 8 | 21.g | even | 6 | 2 | ||
| 49.3.d.b | 8 | 21.h | odd | 6 | 2 | ||
| 441.3.d.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 441.3.d.e | 4 | 7.b | odd | 2 | 1 | inner | |
| 441.3.m.l | 8 | 7.c | even | 3 | 2 | ||
| 441.3.m.l | 8 | 7.d | odd | 6 | 2 | ||
| 784.3.c.c | 4 | 12.b | even | 2 | 1 | ||
| 784.3.c.c | 4 | 84.h | odd | 2 | 1 | ||
| 784.3.s.h | 8 | 84.j | odd | 6 | 2 | ||
| 784.3.s.h | 8 | 84.n | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 1 \)
acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T - 1)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 40T^{2} + 392 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 4 T - 46)^{2} \)
|
| $13$ |
\( T^{4} + 392 T^{2} + 19208 \)
|
| $17$ |
\( T^{4} + 692 T^{2} + 94178 \)
|
| $19$ |
\( T^{4} + 740T^{2} + 4802 \)
|
| $23$ |
\( (T^{2} - 12 T - 92)^{2} \)
|
| $29$ |
\( (T^{2} + 32 T - 136)^{2} \)
|
| $31$ |
\( T^{4} + 1952 T^{2} + 307328 \)
|
| $37$ |
\( (T^{2} - 16 T - 64)^{2} \)
|
| $41$ |
\( T^{4} + 4900 T^{2} + 1387778 \)
|
| $43$ |
\( (T^{2} - 4 T - 878)^{2} \)
|
| $47$ |
\( T^{4} + 1552 T^{2} + 1568 \)
|
| $53$ |
\( (T^{2} - 52 T - 5156)^{2} \)
|
| $59$ |
\( T^{4} + 8980 T^{2} + 14982338 \)
|
| $61$ |
\( T^{4} + 9896 T^{2} + 22767752 \)
|
| $67$ |
\( (T^{2} + 24 T - 1656)^{2} \)
|
| $71$ |
\( (T^{2} - 80 T + 32)^{2} \)
|
| $73$ |
\( T^{4} + 12004 T^{2} + 51842 \)
|
| $79$ |
\( (T^{2} - 184 T + 7496)^{2} \)
|
| $83$ |
\( T^{4} + 7252 T^{2} + 2540258 \)
|
| $89$ |
\( T^{4} + 1300 T^{2} + 61250 \)
|
| $97$ |
\( T^{4} + 9604 T^{2} + 11529602 \)
|
| show more | |
| show less |