Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9898,2,Mod(1,9898)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9898, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9898.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9898 = 2 \cdot 7^{2} \cdot 101 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9898.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: | \(79.0359279207\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.301909.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 5x^{2} + 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1414) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 5x^{2} + 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + \nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} - \beta _1 + 13 \)
|
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 |
|
−1.00000 | −1.87445 | 1.00000 | 3.38803 | 1.87445 | 0 | −1.00000 | 0.513575 | −3.38803 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.64451 | 1.00000 | 2.34892 | 1.64451 | 0 | −1.00000 | −0.295590 | −2.34892 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.601164 | 1.00000 | −1.03744 | 0.601164 | 0 | −1.00000 | −2.63860 | 1.03744 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.37709 | 1.00000 | −1.48071 | −1.37709 | 0 | −1.00000 | −1.10362 | 1.48071 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.74303 | 1.00000 | 2.78120 | −2.74303 | 0 | −1.00000 | 4.52423 | −2.78120 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(101\) | \(1\) |
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 | 9898.2.a.t | 5 | |
| 7.b | odd | 2 | 1 | 1414.2.a.e | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1414.2.a.e | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 9898.2.a.t | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9898))\):
|
\( T_{3}^{5} - 8T_{3}^{3} - 4T_{3}^{2} + 12T_{3} + 7 \)
|
|
\( T_{5}^{5} - 6T_{5}^{4} + 4T_{5}^{3} + 25T_{5}^{2} - 19T_{5} - 34 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} - 8 T^{3} + \cdots + 7 \)
|
| $5$ |
\( T^{5} - 6 T^{4} + \cdots - 34 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + 5 T^{4} + \cdots - 34 \)
|
| $13$ |
\( T^{5} + T^{4} + \cdots + 14 \)
|
| $17$ |
\( T^{5} - 12 T^{4} + \cdots + 284 \)
|
| $19$ |
\( T^{5} - T^{4} + \cdots - 772 \)
|
| $23$ |
\( T^{5} + 4 T^{4} + \cdots + 289 \)
|
| $29$ |
\( T^{5} + 18 T^{4} + \cdots - 1688 \)
|
| $31$ |
\( T^{5} + 8 T^{4} + \cdots + 224 \)
|
| $37$ |
\( T^{5} - 5 T^{4} + \cdots + 2083 \)
|
| $41$ |
\( T^{5} - 13 T^{4} + \cdots + 38 \)
|
| $43$ |
\( T^{5} - T^{4} + \cdots - 24317 \)
|
| $47$ |
\( T^{5} - 152 T^{3} + \cdots - 11276 \)
|
| $53$ |
\( T^{5} + 8 T^{4} + \cdots + 4442 \)
|
| $59$ |
\( T^{5} - 31 T^{4} + \cdots - 821 \)
|
| $61$ |
\( T^{5} - 15 T^{4} + \cdots + 2489 \)
|
| $67$ |
\( T^{5} + 4 T^{4} + \cdots - 7354 \)
|
| $71$ |
\( T^{5} + 21 T^{4} + \cdots - 1853 \)
|
| $73$ |
\( T^{5} - 3 T^{4} + \cdots + 8699 \)
|
| $79$ |
\( T^{5} - T^{4} + \cdots + 353 \)
|
| $83$ |
\( T^{5} - 22 T^{4} + \cdots + 71453 \)
|
| $89$ |
\( T^{5} - 22 T^{4} + \cdots + 25459 \)
|
| $97$ |
\( T^{5} + 4 T^{4} + \cdots - 38726 \)
|
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