Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [280,3,Mod(61,280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(280, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 0, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("280.61");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.be (of order \(6\), degree \(2\), 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: | \(7.62944740209\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/280\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(71\) | \(141\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
−1.00000 | + | 1.73205i | −1.11803 | − | 1.93649i | −2.00000 | − | 3.46410i | 1.11803 | − | 1.93649i | 4.47214 | 6.50000 | − | 2.59808i | 8.00000 | 2.00000 | − | 3.46410i | 2.23607 | + | 3.87298i | ||||||||||||||||
| 61.2 | −1.00000 | + | 1.73205i | 1.11803 | + | 1.93649i | −2.00000 | − | 3.46410i | −1.11803 | + | 1.93649i | −4.47214 | 6.50000 | − | 2.59808i | 8.00000 | 2.00000 | − | 3.46410i | −2.23607 | − | 3.87298i | |||||||||||||||||
| 101.1 | −1.00000 | − | 1.73205i | −1.11803 | + | 1.93649i | −2.00000 | + | 3.46410i | 1.11803 | + | 1.93649i | 4.47214 | 6.50000 | + | 2.59808i | 8.00000 | 2.00000 | + | 3.46410i | 2.23607 | − | 3.87298i | |||||||||||||||||
| 101.2 | −1.00000 | − | 1.73205i | 1.11803 | − | 1.93649i | −2.00000 | + | 3.46410i | −1.11803 | − | 1.93649i | −4.47214 | 6.50000 | + | 2.59808i | 8.00000 | 2.00000 | + | 3.46410i | −2.23607 | + | 3.87298i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.d | odd | 6 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 56.j | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 280.3.be.a | ✓ | 4 |
| 7.d | odd | 6 | 1 | inner | 280.3.be.a | ✓ | 4 |
| 8.b | even | 2 | 1 | inner | 280.3.be.a | ✓ | 4 |
| 56.j | odd | 6 | 1 | inner | 280.3.be.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.3.be.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 280.3.be.a | ✓ | 4 | 7.d | odd | 6 | 1 | inner |
| 280.3.be.a | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 280.3.be.a | ✓ | 4 | 56.j | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 5T_{3}^{2} + 25 \)
acting on \(S_{3}^{\mathrm{new}}(280, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} + 5T^{2} + 25 \)
|
| $5$ |
\( T^{4} + 5T^{2} + 25 \)
|
| $7$ |
\( (T^{2} - 13 T + 49)^{2} \)
|
| $11$ |
\( T^{4} - 240 T^{2} + 57600 \)
|
| $13$ |
\( (T^{2} - 180)^{2} \)
|
| $17$ |
\( (T^{2} + 24 T + 192)^{2} \)
|
| $19$ |
\( T^{4} + 720 T^{2} + 518400 \)
|
| $23$ |
\( (T^{2} - 37 T + 1369)^{2} \)
|
| $29$ |
\( (T^{2} + 15)^{2} \)
|
| $31$ |
\( (T^{2} + 42 T + 588)^{2} \)
|
| $37$ |
\( T^{4} - 540 T^{2} + 291600 \)
|
| $41$ |
\( (T^{2} + 1323)^{2} \)
|
| $43$ |
\( (T^{2} + 5415)^{2} \)
|
| $47$ |
\( (T^{2} + 144 T + 6912)^{2} \)
|
| $53$ |
\( T^{4} - 7260 T^{2} + 52707600 \)
|
| $59$ |
\( T^{4} + 180 T^{2} + 32400 \)
|
| $61$ |
\( T^{4} + 45T^{2} + 2025 \)
|
| $67$ |
\( T^{4} - 7935 T^{2} + 62964225 \)
|
| $71$ |
\( (T - 106)^{4} \)
|
| $73$ |
\( (T^{2} + 30 T + 300)^{2} \)
|
| $79$ |
\( (T^{2} + 64 T + 4096)^{2} \)
|
| $83$ |
\( (T^{2} - 405)^{2} \)
|
| $89$ |
\( (T^{2} - 27 T + 243)^{2} \)
|
| $97$ |
\( (T^{2} + 20172)^{2} \)
|
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