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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.11109.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 15x - 6 \)
|
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 15x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3\nu - 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + \beta _1 + 42 ) / 4 \)
|
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 | −9.40408 | 0 | −1.20950 | 0 | 1.80542 | 0 | 61.4367 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 2.82655 | 0 | 17.4525 | 0 | −4.62596 | 0 | −19.0106 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 8.57753 | 0 | −8.24301 | 0 | 26.8205 | 0 | 46.5740 | 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.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 792.4.a.l | 3 | ||
| 4.b | odd | 2 | 1 | 176.4.a.j | 3 | ||
| 5.b | even | 2 | 1 | 2200.4.a.m | 3 | ||
| 8.b | even | 2 | 1 | 704.4.a.t | 3 | ||
| 8.d | odd | 2 | 1 | 704.4.a.u | 3 | ||
| 11.b | odd | 2 | 1 | 968.4.a.i | 3 | ||
| 12.b | even | 2 | 1 | 1584.4.a.bl | 3 | ||
| 44.c | even | 2 | 1 | 1936.4.a.bh | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.4.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 176.4.a.j | 3 | 4.b | odd | 2 | 1 | ||
| 704.4.a.t | 3 | 8.b | even | 2 | 1 | ||
| 704.4.a.u | 3 | 8.d | odd | 2 | 1 | ||
| 792.4.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 968.4.a.i | 3 | 11.b | odd | 2 | 1 | ||
| 1584.4.a.bl | 3 | 12.b | even | 2 | 1 | ||
| 1936.4.a.bh | 3 | 44.c | even | 2 | 1 | ||
| 2200.4.a.m | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 2T_{3}^{2} - 83T_{3} + 228 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(88))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 2 T^{2} + \cdots + 228 \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots - 174 \)
|
| $7$ |
\( T^{3} - 24 T^{2} + \cdots + 224 \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} - 66 T^{2} + \cdots + 325152 \)
|
| $17$ |
\( T^{3} - 210 T^{2} + \cdots + 70632 \)
|
| $19$ |
\( T^{3} + 72 T^{2} + \cdots - 196672 \)
|
| $23$ |
\( T^{3} + 50 T^{2} + \cdots + 440328 \)
|
| $29$ |
\( T^{3} + 50 T^{2} + \cdots + 1236864 \)
|
| $31$ |
\( T^{3} + 298 T^{2} + \cdots - 14254992 \)
|
| $37$ |
\( T^{3} + 4 T^{2} + \cdots - 11318238 \)
|
| $41$ |
\( T^{3} - 254 T^{2} + \cdots - 971296 \)
|
| $43$ |
\( T^{3} + 112 T^{2} + \cdots - 381456 \)
|
| $47$ |
\( T^{3} - 40 T^{2} + \cdots - 16705536 \)
|
| $53$ |
\( T^{3} + 550 T^{2} + \cdots + 136296 \)
|
| $59$ |
\( T^{3} - 1630 T^{2} + \cdots - 159954212 \)
|
| $61$ |
\( T^{3} + 906 T^{2} + \cdots - 135114784 \)
|
| $67$ |
\( T^{3} + 482 T^{2} + \cdots + 82948068 \)
|
| $71$ |
\( T^{3} - 242 T^{2} + \cdots - 72210312 \)
|
| $73$ |
\( T^{3} - 998 T^{2} + \cdots - 29947936 \)
|
| $79$ |
\( T^{3} - 324 T^{2} + \cdots + 77763136 \)
|
| $83$ |
\( T^{3} - 1360 T^{2} + \cdots + 367318992 \)
|
| $89$ |
\( T^{3} - 1200 T^{2} + \cdots - 7674426 \)
|
| $97$ |
\( T^{3} - 180 T^{2} + \cdots + 97506814 \)
|
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