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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{15}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 15 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = 2\sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−3.74597 | 0 | −113.968 | 80.0807 | 0 | 21.1693 | 906.403 | 0 | −299.980 | ||||||||||||||||||||||||
| 1.2 | 11.7460 | 0 | 9.96773 | 389.919 | 0 | −1249.17 | −1386.40 | 0 | 4579.98 | |||||||||||||||||||||||||
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.c | 2 | |
| 3.b | odd | 2 | 1 | 11.8.a.a | ✓ | 2 | |
| 12.b | even | 2 | 1 | 176.8.a.d | 2 | ||
| 15.d | odd | 2 | 1 | 275.8.a.a | 2 | ||
| 21.c | even | 2 | 1 | 539.8.a.a | 2 | ||
| 33.d | even | 2 | 1 | 121.8.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.8.a.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 99.8.a.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 121.8.a.b | 2 | 33.d | even | 2 | 1 | ||
| 176.8.a.d | 2 | 12.b | even | 2 | 1 | ||
| 275.8.a.a | 2 | 15.d | odd | 2 | 1 | ||
| 539.8.a.a | 2 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 8T_{2} - 44 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 8T - 44 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 470T + 31225 \)
|
| $7$ |
\( T^{2} + 1228T - 26444 \)
|
| $11$ |
\( (T + 1331)^{2} \)
|
| $13$ |
\( T^{2} - 344 T - 16069856 \)
|
| $17$ |
\( T^{2} - 8468 T - 788446604 \)
|
| $19$ |
\( T^{2} + 35280 T - 222369840 \)
|
| $23$ |
\( T^{2} - 61486 T - 159113951 \)
|
| $29$ |
\( T^{2} + 179040 T + 122928960 \)
|
| $31$ |
\( T^{2} + \cdots - 23689658111 \)
|
| $37$ |
\( T^{2} + \cdots + 191745594561 \)
|
| $41$ |
\( T^{2} + \cdots - 264475105136 \)
|
| $43$ |
\( T^{2} + \cdots + 9203802564 \)
|
| $47$ |
\( T^{2} + \cdots + 677029127296 \)
|
| $53$ |
\( T^{2} + \cdots + 388813137924 \)
|
| $59$ |
\( T^{2} + \cdots + 207772229185 \)
|
| $61$ |
\( T^{2} + \cdots + 9007507329744 \)
|
| $67$ |
\( T^{2} + \cdots - 9239984638199 \)
|
| $71$ |
\( T^{2} + \cdots - 11975092638711 \)
|
| $73$ |
\( T^{2} + \cdots - 2090754692576 \)
|
| $79$ |
\( T^{2} + \cdots - 2681742596060 \)
|
| $83$ |
\( T^{2} + \cdots - 536001911676 \)
|
| $89$ |
\( T^{2} + \cdots + 42998937114225 \)
|
| $97$ |
\( T^{2} + \cdots - 1834997592239 \)
|
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