Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4732,2,Mod(337,4732)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4732.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4732.g (of order \(2\), degree \(1\), not 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: | \(37.7852102365\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 364) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{12}^{2} - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{3} + 2\zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( \beta_{2} + 1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times\).
| \(n\) | \(2367\) | \(2705\) | \(4565\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −0.732051 | 0 | − | 1.73205i | 0 | 1.00000i | 0 | −2.46410 | 0 | |||||||||||||||||||||||||||||
| 337.2 | 0 | −0.732051 | 0 | 1.73205i | 0 | − | 1.00000i | 0 | −2.46410 | 0 | ||||||||||||||||||||||||||||||
| 337.3 | 0 | 2.73205 | 0 | − | 1.73205i | 0 | − | 1.00000i | 0 | 4.46410 | 0 | |||||||||||||||||||||||||||||
| 337.4 | 0 | 2.73205 | 0 | 1.73205i | 0 | 1.00000i | 0 | 4.46410 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4732.2.g.h | 4 | |
| 13.b | even | 2 | 1 | inner | 4732.2.g.h | 4 | |
| 13.d | odd | 4 | 1 | 364.2.a.d | ✓ | 2 | |
| 13.d | odd | 4 | 1 | 4732.2.a.l | 2 | ||
| 39.f | even | 4 | 1 | 3276.2.a.n | 2 | ||
| 52.f | even | 4 | 1 | 1456.2.a.n | 2 | ||
| 65.g | odd | 4 | 1 | 9100.2.a.o | 2 | ||
| 91.i | even | 4 | 1 | 2548.2.a.l | 2 | ||
| 91.z | odd | 12 | 2 | 2548.2.j.k | 4 | ||
| 91.bb | even | 12 | 2 | 2548.2.j.n | 4 | ||
| 104.j | odd | 4 | 1 | 5824.2.a.bh | 2 | ||
| 104.m | even | 4 | 1 | 5824.2.a.bq | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 364.2.a.d | ✓ | 2 | 13.d | odd | 4 | 1 | |
| 1456.2.a.n | 2 | 52.f | even | 4 | 1 | ||
| 2548.2.a.l | 2 | 91.i | even | 4 | 1 | ||
| 2548.2.j.k | 4 | 91.z | odd | 12 | 2 | ||
| 2548.2.j.n | 4 | 91.bb | even | 12 | 2 | ||
| 3276.2.a.n | 2 | 39.f | even | 4 | 1 | ||
| 4732.2.a.l | 2 | 13.d | odd | 4 | 1 | ||
| 4732.2.g.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 4732.2.g.h | 4 | 13.b | even | 2 | 1 | inner | |
| 5824.2.a.bh | 2 | 104.j | odd | 4 | 1 | ||
| 5824.2.a.bq | 2 | 104.m | even | 4 | 1 | ||
| 9100.2.a.o | 2 | 65.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4732, [\chi])\):
|
\( T_{3}^{2} - 2T_{3} - 2 \)
|
|
\( T_{5}^{2} + 3 \)
|
|
\( T_{17}^{2} + 6T_{17} - 18 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 2 T - 2)^{2} \)
|
| $5$ |
\( (T^{2} + 3)^{2} \)
|
| $7$ |
\( (T^{2} + 1)^{2} \)
|
| $11$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 6 T - 18)^{2} \)
|
| $19$ |
\( T^{4} + 62T^{2} + 529 \)
|
| $23$ |
\( (T + 3)^{4} \)
|
| $29$ |
\( (T^{2} + 6 T - 3)^{2} \)
|
| $31$ |
\( T^{4} + 86T^{2} + 121 \)
|
| $37$ |
\( T^{4} + 56T^{2} + 676 \)
|
| $41$ |
\( (T^{2} + 12)^{2} \)
|
| $43$ |
\( (T - 7)^{4} \)
|
| $47$ |
\( T^{4} + 78T^{2} + 1089 \)
|
| $53$ |
\( (T^{2} + 6 T - 39)^{2} \)
|
| $59$ |
\( (T^{2} + 108)^{2} \)
|
| $61$ |
\( (T + 10)^{4} \)
|
| $67$ |
\( T^{4} + 248T^{2} + 8464 \)
|
| $71$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $73$ |
\( T^{4} + 86T^{2} + 121 \)
|
| $79$ |
\( (T^{2} + 2 T - 107)^{2} \)
|
| $83$ |
\( T^{4} + 126T^{2} + 81 \)
|
| $89$ |
\( T^{4} + 126T^{2} + 81 \)
|
| $97$ |
\( T^{4} + 182T^{2} + 1369 \)
|
| show more | |
| show less |