Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [387,2,Mod(1,387)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(387, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("387.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 387 = 3^{2} \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 387.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: | \(3.09021055822\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{5}, \sqrt{13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - 9x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 9\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 9\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.23607 | 0 | 3.00000 | −2.92081 | 0 | −4.53113 | −2.23607 | 0 | 6.53113 | ||||||||||||||||||||||||||||||
| 1.2 | −2.23607 | 0 | 3.00000 | 0.684742 | 0 | 3.53113 | −2.23607 | 0 | −1.53113 | |||||||||||||||||||||||||||||||
| 1.3 | 2.23607 | 0 | 3.00000 | −0.684742 | 0 | 3.53113 | 2.23607 | 0 | −1.53113 | |||||||||||||||||||||||||||||||
| 1.4 | 2.23607 | 0 | 3.00000 | 2.92081 | 0 | −4.53113 | 2.23607 | 0 | 6.53113 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(43\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 387.2.a.j | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 387.2.a.j | ✓ | 4 |
| 4.b | odd | 2 | 1 | 6192.2.a.bz | 4 | ||
| 5.b | even | 2 | 1 | 9675.2.a.bv | 4 | ||
| 12.b | even | 2 | 1 | 6192.2.a.bz | 4 | ||
| 15.d | odd | 2 | 1 | 9675.2.a.bv | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 387.2.a.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 387.2.a.j | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 6192.2.a.bz | 4 | 4.b | odd | 2 | 1 | ||
| 6192.2.a.bz | 4 | 12.b | even | 2 | 1 | ||
| 9675.2.a.bv | 4 | 5.b | even | 2 | 1 | ||
| 9675.2.a.bv | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(387))\):
|
\( T_{2}^{2} - 5 \)
|
|
\( T_{5}^{4} - 9T_{5}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 5)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 9T^{2} + 4 \)
|
| $7$ |
\( (T^{2} + T - 16)^{2} \)
|
| $11$ |
\( T^{4} - 9T^{2} + 4 \)
|
| $13$ |
\( (T^{2} - 5 T - 10)^{2} \)
|
| $17$ |
\( T^{4} - 36T^{2} + 64 \)
|
| $19$ |
\( (T^{2} + T - 16)^{2} \)
|
| $23$ |
\( T^{4} - 36T^{2} + 64 \)
|
| $29$ |
\( T^{4} - 81T^{2} + 324 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T - 6)^{4} \)
|
| $41$ |
\( T^{4} - 144T^{2} + 1024 \)
|
| $43$ |
\( (T - 1)^{4} \)
|
| $47$ |
\( T^{4} - 81T^{2} + 324 \)
|
| $53$ |
\( T^{4} - 144T^{2} + 1024 \)
|
| $59$ |
\( T^{4} - 36T^{2} + 64 \)
|
| $61$ |
\( (T + 2)^{4} \)
|
| $67$ |
\( (T - 4)^{4} \)
|
| $71$ |
\( T^{4} - 244 T^{2} + 12544 \)
|
| $73$ |
\( (T - 6)^{4} \)
|
| $79$ |
\( (T^{2} + 14 T - 16)^{2} \)
|
| $83$ |
\( T^{4} - 121T^{2} + 4 \)
|
| $89$ |
\( (T^{2} - 180)^{2} \)
|
| $97$ |
\( (T^{2} - 19 T + 74)^{2} \)
|
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