Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,6,Mod(1,99)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(99, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 99.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: | \(15.8779981615\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.54492.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 52x - 38 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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
\( x^{3} - x^{2} - 52x - 38 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 34 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + \beta _1 + 35 \)
|
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 |
|
−8.18772 | 0 | 35.0388 | 59.8722 | 0 | 145.071 | −24.8808 | 0 | −490.217 | |||||||||||||||||||||||||||
| 1.2 | −2.20859 | 0 | −27.1221 | −75.2230 | 0 | −225.525 | 130.577 | 0 | 166.137 | ||||||||||||||||||||||||||||
| 1.3 | 10.3963 | 0 | 76.0833 | −8.64919 | 0 | 164.454 | 458.304 | 0 | −89.9197 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(11\) | \(1\) |
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 | 99.6.a.g | 3 | |
| 3.b | odd | 2 | 1 | 11.6.a.b | ✓ | 3 | |
| 11.b | odd | 2 | 1 | 1089.6.a.r | 3 | ||
| 12.b | even | 2 | 1 | 176.6.a.i | 3 | ||
| 15.d | odd | 2 | 1 | 275.6.a.b | 3 | ||
| 15.e | even | 4 | 2 | 275.6.b.b | 6 | ||
| 21.c | even | 2 | 1 | 539.6.a.e | 3 | ||
| 24.f | even | 2 | 1 | 704.6.a.t | 3 | ||
| 24.h | odd | 2 | 1 | 704.6.a.q | 3 | ||
| 33.d | even | 2 | 1 | 121.6.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.6.a.b | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 99.6.a.g | 3 | 1.a | even | 1 | 1 | trivial | |
| 121.6.a.d | 3 | 33.d | even | 2 | 1 | ||
| 176.6.a.i | 3 | 12.b | even | 2 | 1 | ||
| 275.6.a.b | 3 | 15.d | odd | 2 | 1 | ||
| 275.6.b.b | 6 | 15.e | even | 4 | 2 | ||
| 539.6.a.e | 3 | 21.c | even | 2 | 1 | ||
| 704.6.a.q | 3 | 24.h | odd | 2 | 1 | ||
| 704.6.a.t | 3 | 24.f | even | 2 | 1 | ||
| 1089.6.a.r | 3 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 90T_{2} - 188 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 90T - 188 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 24 T^{2} + \cdots - 38954 \)
|
| $7$ |
\( T^{3} - 84 T^{2} + \cdots + 5380448 \)
|
| $11$ |
\( (T + 121)^{3} \)
|
| $13$ |
\( T^{3} - 486 T^{2} + \cdots + 164136608 \)
|
| $17$ |
\( T^{3} + 1086 T^{2} + \cdots - 331752056 \)
|
| $19$ |
\( T^{3} - 1380 T^{2} + \cdots + 57024000 \)
|
| $23$ |
\( T^{3} + \cdots + 17004325928 \)
|
| $29$ |
\( T^{3} + \cdots + 4029189120 \)
|
| $31$ |
\( T^{3} + \cdots + 1094344400 \)
|
| $37$ |
\( T^{3} + \cdots + 541788167034 \)
|
| $41$ |
\( T^{3} + \cdots - 201929821568 \)
|
| $43$ |
\( T^{3} + \cdots - 2443875098544 \)
|
| $47$ |
\( T^{3} + \cdots + 70174939136 \)
|
| $53$ |
\( T^{3} + \cdots + 1850911309656 \)
|
| $59$ |
\( T^{3} + \cdots - 7759637437060 \)
|
| $61$ |
\( T^{3} + \cdots + 15233874751008 \)
|
| $67$ |
\( T^{3} + \cdots - 147288561330212 \)
|
| $71$ |
\( T^{3} + \cdots + 1290398551704 \)
|
| $73$ |
\( T^{3} + \cdots - 34539701265952 \)
|
| $79$ |
\( T^{3} + \cdots + 1279883216320 \)
|
| $83$ |
\( T^{3} + \cdots + 411597824719824 \)
|
| $89$ |
\( T^{3} + \cdots + 90320980174650 \)
|
| $97$ |
\( T^{3} + \cdots - 10221902527106 \)
|
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