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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{33}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). 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 |
|
−9.37228 | 0 | 55.8397 | −0.277187 | 0 | −105.081 | −223.432 | 0 | 2.59787 | ||||||||||||||||||||||||
| 1.2 | −3.62772 | 0 | −18.8397 | −57.7228 | 0 | 251.081 | 184.432 | 0 | 209.402 | |||||||||||||||||||||||||
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.d | 2 | |
| 3.b | odd | 2 | 1 | 33.6.a.e | ✓ | 2 | |
| 11.b | odd | 2 | 1 | 1089.6.a.p | 2 | ||
| 12.b | even | 2 | 1 | 528.6.a.o | 2 | ||
| 15.d | odd | 2 | 1 | 825.6.a.c | 2 | ||
| 33.d | even | 2 | 1 | 363.6.a.f | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.6.a.e | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 99.6.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 363.6.a.f | 2 | 33.d | even | 2 | 1 | ||
| 528.6.a.o | 2 | 12.b | even | 2 | 1 | ||
| 825.6.a.c | 2 | 15.d | odd | 2 | 1 | ||
| 1089.6.a.p | 2 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 13T_{2} + 34 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 13T + 34 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 58T + 16 \)
|
| $7$ |
\( T^{2} - 146T - 26384 \)
|
| $11$ |
\( (T - 121)^{2} \)
|
| $13$ |
\( T^{2} + 130T - 40952 \)
|
| $17$ |
\( T^{2} - 728 T - 1009172 \)
|
| $19$ |
\( T^{2} + 828 T - 5817312 \)
|
| $23$ |
\( T^{2} - 238T - 884264 \)
|
| $29$ |
\( T^{2} + 696 T - 18243924 \)
|
| $31$ |
\( T^{2} + 10480 T + 6337600 \)
|
| $37$ |
\( T^{2} + 1908 T - 64511196 \)
|
| $41$ |
\( T^{2} + 36484 T + 332770036 \)
|
| $43$ |
\( T^{2} - 9768 T - 56044032 \)
|
| $47$ |
\( T^{2} + 43742 T + 477085816 \)
|
| $53$ |
\( T^{2} - 12174 T - 379065264 \)
|
| $59$ |
\( T^{2} - 2788 T - 30400064 \)
|
| $61$ |
\( T^{2} + 25302 T + 126184224 \)
|
| $67$ |
\( T^{2} + 40520 T - 921013232 \)
|
| $71$ |
\( T^{2} + 31386 T + 89872392 \)
|
| $73$ |
\( T^{2} + 46780 T + 291320452 \)
|
| $79$ |
\( T^{2} + \cdots - 1006085552 \)
|
| $83$ |
\( T^{2} + \cdots - 2400814800 \)
|
| $89$ |
\( T^{2} + \cdots - 5148586428 \)
|
| $97$ |
\( T^{2} + \cdots + 13776267004 \)
|
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