Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9000,2,Mod(1,9000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9000, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9000.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9000 = 2^{3} \cdot 3^{2} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9000.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: | \(71.8653618192\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 3000) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). 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 |
|
0 | 0 | 0 | 0 | 0 | 1.38197 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0 | 0 | 3.61803 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 9000.2.a.p | 2 | |
| 3.b | odd | 2 | 1 | 3000.2.a.d | ✓ | 2 | |
| 5.b | even | 2 | 1 | 9000.2.a.a | 2 | ||
| 12.b | even | 2 | 1 | 6000.2.a.p | 2 | ||
| 15.d | odd | 2 | 1 | 3000.2.a.e | yes | 2 | |
| 15.e | even | 4 | 2 | 3000.2.f.c | 4 | ||
| 60.h | even | 2 | 1 | 6000.2.a.l | 2 | ||
| 60.l | odd | 4 | 2 | 6000.2.f.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3000.2.a.d | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 3000.2.a.e | yes | 2 | 15.d | odd | 2 | 1 | |
| 3000.2.f.c | 4 | 15.e | even | 4 | 2 | ||
| 6000.2.a.l | 2 | 60.h | even | 2 | 1 | ||
| 6000.2.a.p | 2 | 12.b | even | 2 | 1 | ||
| 6000.2.f.i | 4 | 60.l | odd | 4 | 2 | ||
| 9000.2.a.a | 2 | 5.b | even | 2 | 1 | ||
| 9000.2.a.p | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9000))\):
|
\( T_{7}^{2} - 5T_{7} + 5 \)
|
|
\( T_{11} - 1 \)
|
|
\( T_{13}^{2} + 4T_{13} - 1 \)
|
|
\( T_{17}^{2} - T_{17} - 11 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} - 5T + 5 \)
|
| $11$ |
\( (T - 1)^{2} \)
|
| $13$ |
\( T^{2} + 4T - 1 \)
|
| $17$ |
\( T^{2} - T - 11 \)
|
| $19$ |
\( T^{2} + 8T + 11 \)
|
| $23$ |
\( T^{2} + 4T - 41 \)
|
| $29$ |
\( T^{2} - 8T + 11 \)
|
| $31$ |
\( T^{2} + 9T - 11 \)
|
| $37$ |
\( T^{2} + 8T + 11 \)
|
| $41$ |
\( T^{2} - 7T - 19 \)
|
| $43$ |
\( T^{2} - 13T + 31 \)
|
| $47$ |
\( T^{2} - 5 \)
|
| $53$ |
\( T^{2} - 9T + 19 \)
|
| $59$ |
\( T^{2} - 17T + 61 \)
|
| $61$ |
\( T^{2} - T - 11 \)
|
| $67$ |
\( T^{2} - 12T + 16 \)
|
| $71$ |
\( T^{2} + T - 101 \)
|
| $73$ |
\( T^{2} - 5T - 5 \)
|
| $79$ |
\( T^{2} - 14T + 4 \)
|
| $83$ |
\( T^{2} - 13T + 11 \)
|
| $89$ |
\( (T - 11)^{2} \)
|
| $97$ |
\( T^{2} - 9T - 131 \)
|
| show more | |
| show less |