Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(76.0659748754\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.202533.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 37x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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
\( x^{3} - x^{2} - 37x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu^{2} + 6\nu - 52 \)
|
| \(\beta_{2}\) | \(=\) |
\( -4\nu^{2} + 12\nu + 96 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 8 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 2\beta _1 + 200 ) / 8 \)
|
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 |
|
−52.3190 | 0 | 689.274 | −913.580 | 0 | −59861.9 | 71087.1 | 0 | 47797.6 | |||||||||||||||||||||||||||
| 1.2 | −23.3081 | 0 | −1504.73 | 13329.8 | 0 | 32948.8 | 82807.5 | 0 | −310692. | ||||||||||||||||||||||||||||
| 1.3 | 75.6271 | 0 | 3671.46 | −5111.20 | 0 | 21831.1 | 122777. | 0 | −386545. | ||||||||||||||||||||||||||||
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 | 99.12.a.a | 3 | |
| 3.b | odd | 2 | 1 | 11.12.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 176.12.a.e | 3 | ||
| 15.d | odd | 2 | 1 | 275.12.a.a | 3 | ||
| 33.d | even | 2 | 1 | 121.12.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.12.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 99.12.a.a | 3 | 1.a | even | 1 | 1 | trivial | |
| 121.12.a.c | 3 | 33.d | even | 2 | 1 | ||
| 176.12.a.e | 3 | 12.b | even | 2 | 1 | ||
| 275.12.a.a | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4500T_{2} - 92224 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 4500T - 92224 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 62243294875 \)
|
| $7$ |
\( T^{3} + \cdots + 43059126844184 \)
|
| $11$ |
\( (T + 161051)^{3} \)
|
| $13$ |
\( T^{3} + \cdots + 33\!\cdots\!56 \)
|
| $17$ |
\( T^{3} + \cdots - 21\!\cdots\!48 \)
|
| $19$ |
\( T^{3} + \cdots - 16\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 14\!\cdots\!59 \)
|
| $29$ |
\( T^{3} + \cdots - 18\!\cdots\!40 \)
|
| $31$ |
\( T^{3} + \cdots + 69\!\cdots\!75 \)
|
| $37$ |
\( T^{3} + \cdots - 98\!\cdots\!23 \)
|
| $41$ |
\( T^{3} + \cdots - 60\!\cdots\!08 \)
|
| $43$ |
\( T^{3} + \cdots + 19\!\cdots\!72 \)
|
| $47$ |
\( T^{3} + \cdots + 45\!\cdots\!28 \)
|
| $53$ |
\( T^{3} + \cdots + 30\!\cdots\!72 \)
|
| $59$ |
\( T^{3} + \cdots + 26\!\cdots\!15 \)
|
| $61$ |
\( T^{3} + \cdots + 79\!\cdots\!68 \)
|
| $67$ |
\( T^{3} + \cdots + 86\!\cdots\!09 \)
|
| $71$ |
\( T^{3} + \cdots + 11\!\cdots\!39 \)
|
| $73$ |
\( T^{3} + \cdots + 49\!\cdots\!76 \)
|
| $79$ |
\( T^{3} + \cdots + 20\!\cdots\!20 \)
|
| $83$ |
\( T^{3} + \cdots - 54\!\cdots\!32 \)
|
| $89$ |
\( T^{3} + \cdots + 22\!\cdots\!15 \)
|
| $97$ |
\( T^{3} + \cdots - 28\!\cdots\!93 \)
|
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