Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [88,4,Mod(1,88)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(88, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("88.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 88 = 2^{3} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 88.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: | \(5.19216808051\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
0 | −5.47214 | 0 | 14.8885 | 0 | −32.4721 | 0 | 2.94427 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 3.47214 | 0 | −20.8885 | 0 | −23.5279 | 0 | −14.9443 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 88.4.a.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 792.4.a.g | 2 | ||
| 4.b | odd | 2 | 1 | 176.4.a.h | 2 | ||
| 5.b | even | 2 | 1 | 2200.4.a.k | 2 | ||
| 8.b | even | 2 | 1 | 704.4.a.q | 2 | ||
| 8.d | odd | 2 | 1 | 704.4.a.o | 2 | ||
| 11.b | odd | 2 | 1 | 968.4.a.f | 2 | ||
| 12.b | even | 2 | 1 | 1584.4.a.bg | 2 | ||
| 44.c | even | 2 | 1 | 1936.4.a.t | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 176.4.a.h | 2 | 4.b | odd | 2 | 1 | ||
| 704.4.a.o | 2 | 8.d | odd | 2 | 1 | ||
| 704.4.a.q | 2 | 8.b | even | 2 | 1 | ||
| 792.4.a.g | 2 | 3.b | odd | 2 | 1 | ||
| 968.4.a.f | 2 | 11.b | odd | 2 | 1 | ||
| 1584.4.a.bg | 2 | 12.b | even | 2 | 1 | ||
| 1936.4.a.t | 2 | 44.c | even | 2 | 1 | ||
| 2200.4.a.k | 2 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 2T_{3} - 19 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(88))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 2T - 19 \)
|
| $5$ |
\( T^{2} + 6T - 311 \)
|
| $7$ |
\( T^{2} + 56T + 764 \)
|
| $11$ |
\( (T - 11)^{2} \)
|
| $13$ |
\( T^{2} + 44T - 1936 \)
|
| $17$ |
\( T^{2} + 8T - 4484 \)
|
| $19$ |
\( T^{2} + 188T + 7216 \)
|
| $23$ |
\( T^{2} + 66T - 15731 \)
|
| $29$ |
\( T^{2} + 56T - 41536 \)
|
| $31$ |
\( T^{2} - 206T + 9629 \)
|
| $37$ |
\( T^{2} - 142T - 49039 \)
|
| $41$ |
\( T^{2} - 252T + 12496 \)
|
| $43$ |
\( T^{2} + 372T + 18916 \)
|
| $47$ |
\( T^{2} + 200T + 3520 \)
|
| $53$ |
\( T^{2} + 1060 T + 276980 \)
|
| $59$ |
\( T^{2} + 782T + 108701 \)
|
| $61$ |
\( T^{2} + 240T - 83600 \)
|
| $67$ |
\( T^{2} - 282T - 64619 \)
|
| $71$ |
\( T^{2} - 754T + 137629 \)
|
| $73$ |
\( T^{2} - 68T - 517264 \)
|
| $79$ |
\( T^{2} - 52T - 154204 \)
|
| $83$ |
\( T^{2} - 4T - 551116 \)
|
| $89$ |
\( T^{2} - 278T - 672599 \)
|
| $97$ |
\( T^{2} - 306T - 192911 \)
|
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