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: | \(\Q(i, \sqrt{13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 7x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 in terms of a root \(\nu\) of
\( x^{4} + 7x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{2} - 4\beta_1 \)
|
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 | −1.30278 | 0 | − | 3.30278i | 0 | − | 1.00000i | 0 | −1.30278 | 0 | ||||||||||||||||||||||||||||
| 337.2 | 0 | −1.30278 | 0 | 3.30278i | 0 | 1.00000i | 0 | −1.30278 | 0 | |||||||||||||||||||||||||||||||
| 337.3 | 0 | 2.30278 | 0 | − | 0.302776i | 0 | 1.00000i | 0 | 2.30278 | 0 | ||||||||||||||||||||||||||||||
| 337.4 | 0 | 2.30278 | 0 | 0.302776i | 0 | − | 1.00000i | 0 | 2.30278 | 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.g | 4 | |
| 13.b | even | 2 | 1 | inner | 4732.2.g.g | 4 | |
| 13.d | odd | 4 | 1 | 4732.2.a.j | 2 | ||
| 13.d | odd | 4 | 1 | 4732.2.a.k | 2 | ||
| 13.f | odd | 12 | 2 | 364.2.k.c | ✓ | 4 | |
| 39.k | even | 12 | 2 | 3276.2.z.d | 4 | ||
| 52.l | even | 12 | 2 | 1456.2.s.m | 4 | ||
| 91.w | even | 12 | 2 | 2548.2.l.i | 4 | ||
| 91.x | odd | 12 | 2 | 2548.2.i.j | 4 | ||
| 91.ba | even | 12 | 2 | 2548.2.i.l | 4 | ||
| 91.bc | even | 12 | 2 | 2548.2.k.f | 4 | ||
| 91.bd | odd | 12 | 2 | 2548.2.l.k | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 364.2.k.c | ✓ | 4 | 13.f | odd | 12 | 2 | |
| 1456.2.s.m | 4 | 52.l | even | 12 | 2 | ||
| 2548.2.i.j | 4 | 91.x | odd | 12 | 2 | ||
| 2548.2.i.l | 4 | 91.ba | even | 12 | 2 | ||
| 2548.2.k.f | 4 | 91.bc | even | 12 | 2 | ||
| 2548.2.l.i | 4 | 91.w | even | 12 | 2 | ||
| 2548.2.l.k | 4 | 91.bd | odd | 12 | 2 | ||
| 3276.2.z.d | 4 | 39.k | even | 12 | 2 | ||
| 4732.2.a.j | 2 | 13.d | odd | 4 | 1 | ||
| 4732.2.a.k | 2 | 13.d | odd | 4 | 1 | ||
| 4732.2.g.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 4732.2.g.g | 4 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4732, [\chi])\):
|
\( T_{3}^{2} - T_{3} - 3 \)
|
|
\( T_{5}^{4} + 11T_{5}^{2} + 1 \)
|
|
\( T_{17}^{2} + 4T_{17} - 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - T - 3)^{2} \)
|
| $5$ |
\( T^{4} + 11T^{2} + 1 \)
|
| $7$ |
\( (T^{2} + 1)^{2} \)
|
| $11$ |
\( T^{4} + 47T^{2} + 289 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 4 T - 9)^{2} \)
|
| $19$ |
\( T^{4} + 67T^{2} + 729 \)
|
| $23$ |
\( (T + 2)^{4} \)
|
| $29$ |
\( (T^{2} - 3 T - 1)^{2} \)
|
| $31$ |
\( T^{4} + 58T^{2} + 9 \)
|
| $37$ |
\( T^{4} + 112T^{2} + 2304 \)
|
| $41$ |
\( T^{4} + 76T^{2} + 144 \)
|
| $43$ |
\( (T^{2} - T - 29)^{2} \)
|
| $47$ |
\( T^{4} + 154T^{2} + 2601 \)
|
| $53$ |
\( (T^{2} - 117)^{2} \)
|
| $59$ |
\( T^{4} + 34T^{2} + 81 \)
|
| $61$ |
\( (T^{2} - 52)^{2} \)
|
| $67$ |
\( (T^{2} + 169)^{2} \)
|
| $71$ |
\( T^{4} + 236 T^{2} + 13456 \)
|
| $73$ |
\( T^{4} + 136T^{2} + 1296 \)
|
| $79$ |
\( (T - 8)^{4} \)
|
| $83$ |
\( T^{4} + 154T^{2} + 2601 \)
|
| $89$ |
\( T^{4} + 279T^{2} + 6561 \)
|
| $97$ |
\( T^{4} + 323 T^{2} + 10609 \)
|
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