Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [89,2,Mod(1,89)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(89, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("89.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 89.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: | \(0.710668577989\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.535120.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 4x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 6x^{3} + 4x^{2} + 4x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 2\nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 2\nu^{3} + 6\nu^{2} - 3\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 8\nu^{2} - \nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 2\beta_{2} + 6\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 11\beta_{3} + 10\beta_{2} + 15\beta _1 + 24 \)
|
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 |
|
−2.47725 | −2.75935 | 4.13678 | −2.13678 | 6.83562 | 3.26017 | −5.29335 | 4.61403 | 5.29335 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.09813 | 2.34526 | 2.40214 | −0.402140 | −4.92067 | 0.678986 | −0.843741 | 2.50027 | 0.843741 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.745954 | −0.549910 | −1.44355 | 3.44355 | 0.410207 | 5.17498 | 2.56873 | −2.69760 | −2.56873 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.62791 | −0.148907 | 0.650080 | 1.34992 | −0.242407 | −2.21676 | −2.19754 | −2.97783 | 2.19754 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.69343 | −1.88709 | 5.25455 | −3.25455 | −5.08275 | 1.10263 | 8.76590 | 0.561124 | −8.76590 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(89\) | \(-1\) |
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 | 89.2.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 801.2.a.i | 5 | ||
| 4.b | odd | 2 | 1 | 1424.2.a.k | 5 | ||
| 5.b | even | 2 | 1 | 2225.2.a.j | 5 | ||
| 7.b | odd | 2 | 1 | 4361.2.a.i | 5 | ||
| 8.b | even | 2 | 1 | 5696.2.a.be | 5 | ||
| 8.d | odd | 2 | 1 | 5696.2.a.bc | 5 | ||
| 89.b | even | 2 | 1 | 7921.2.a.g | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 89.2.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 801.2.a.i | 5 | 3.b | odd | 2 | 1 | ||
| 1424.2.a.k | 5 | 4.b | odd | 2 | 1 | ||
| 2225.2.a.j | 5 | 5.b | even | 2 | 1 | ||
| 4361.2.a.i | 5 | 7.b | odd | 2 | 1 | ||
| 5696.2.a.bc | 5 | 8.d | odd | 2 | 1 | ||
| 5696.2.a.be | 5 | 8.b | even | 2 | 1 | ||
| 7921.2.a.g | 5 | 89.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + T_{2}^{4} - 10T_{2}^{3} - 10T_{2}^{2} + 21T_{2} + 17 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(89))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} + \cdots + 17 \)
|
| $3$ |
\( T^{5} + 3 T^{4} + \cdots - 1 \)
|
| $5$ |
\( T^{5} + T^{4} + \cdots + 13 \)
|
| $7$ |
\( T^{5} - 8 T^{4} + \cdots + 28 \)
|
| $11$ |
\( T^{5} - 6 T^{4} + \cdots - 112 \)
|
| $13$ |
\( T^{5} - 28 T^{3} + \cdots + 16 \)
|
| $17$ |
\( T^{5} + 13 T^{4} + \cdots - 883 \)
|
| $19$ |
\( T^{5} - 13 T^{4} + \cdots + 199 \)
|
| $23$ |
\( T^{5} - T^{4} + \cdots - 1657 \)
|
| $29$ |
\( T^{5} - 2 T^{4} + \cdots - 784 \)
|
| $31$ |
\( T^{5} - 19 T^{4} + \cdots + 7 \)
|
| $37$ |
\( T^{5} + 14 T^{4} + \cdots + 1120 \)
|
| $41$ |
\( T^{5} + 2 T^{4} + \cdots - 1072 \)
|
| $43$ |
\( T^{5} - T^{4} + \cdots + 1573 \)
|
| $47$ |
\( T^{5} + 4 T^{4} + \cdots - 16 \)
|
| $53$ |
\( T^{5} + 11 T^{4} + \cdots + 1319 \)
|
| $59$ |
\( T^{5} - 118 T^{3} + \cdots + 1580 \)
|
| $61$ |
\( T^{5} - 4 T^{4} + \cdots - 16 \)
|
| $67$ |
\( T^{5} - 4 T^{4} + \cdots + 2000 \)
|
| $71$ |
\( T^{5} + 2 T^{4} + \cdots + 47008 \)
|
| $73$ |
\( T^{5} + 25 T^{4} + \cdots - 3475 \)
|
| $79$ |
\( T^{5} - 54 T^{4} + \cdots - 74464 \)
|
| $83$ |
\( T^{5} + 20 T^{4} + \cdots + 196 \)
|
| $89$ |
\( (T - 1)^{5} \)
|
| $97$ |
\( T^{5} - 13 T^{4} + \cdots + 21599 \)
|
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