Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [99,8,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, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("99.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| 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: | \(30.9261175229\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 510x^{2} - 1544x + 28880 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 510x^{2} - 1544x + 28880 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6\nu - 254 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 11\nu^{2} - 340\nu + 1352 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6\beta _1 + 254 \)
|
| \(\nu^{3}\) | \(=\) |
\( 6\beta_{3} + 11\beta_{2} + 406\beta _1 + 1442 \)
|
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 |
|
−21.6807 | 0 | 342.055 | 369.874 | 0 | 1000.53 | −4640.87 | 0 | −8019.14 | ||||||||||||||||||||||||||||||
| 1.2 | −15.0316 | 0 | 97.9505 | −367.278 | 0 | −91.9512 | 451.694 | 0 | 5520.80 | |||||||||||||||||||||||||||||||
| 1.3 | 2.33327 | 0 | −122.556 | 27.4165 | 0 | 860.612 | −584.614 | 0 | 63.9699 | |||||||||||||||||||||||||||||||
| 1.4 | 19.3791 | 0 | 247.551 | −336.012 | 0 | −879.192 | 2316.79 | 0 | −6511.62 | |||||||||||||||||||||||||||||||
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 | 99.8.a.f | 4 | |
| 3.b | odd | 2 | 1 | 33.8.a.e | ✓ | 4 | |
| 12.b | even | 2 | 1 | 528.8.a.r | 4 | ||
| 33.d | even | 2 | 1 | 363.8.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.8.a.e | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 99.8.a.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 363.8.a.f | 4 | 33.d | even | 2 | 1 | ||
| 528.8.a.r | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 15T_{2}^{3} - 426T_{2}^{2} - 5416T_{2} + 14736 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 15 T^{3} + \cdots + 14736 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 1251456000 \)
|
| $7$ |
\( T^{4} + \cdots + 69611139776 \)
|
| $11$ |
\( (T + 1331)^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 161846909982208 \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots + 43\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots - 49\!\cdots\!52 \)
|
| $29$ |
\( T^{4} + \cdots + 66\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots + 59\!\cdots\!52 \)
|
| $37$ |
\( T^{4} + \cdots + 37\!\cdots\!72 \)
|
| $41$ |
\( T^{4} + \cdots - 49\!\cdots\!64 \)
|
| $43$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 25\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 80\!\cdots\!56 \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots - 13\!\cdots\!48 \)
|
| $67$ |
\( T^{4} + \cdots - 15\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots - 50\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots + 45\!\cdots\!12 \)
|
| $79$ |
\( T^{4} + \cdots - 88\!\cdots\!40 \)
|
| $83$ |
\( T^{4} + \cdots + 11\!\cdots\!72 \)
|
| $89$ |
\( T^{4} + \cdots - 11\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots - 11\!\cdots\!36 \)
|
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