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: | \(1\) |
| 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: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 | −1.23607 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0 | 0 | 3.23607 | 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.m | yes | 2 |
| 3.b | odd | 2 | 1 | 9000.2.a.l | yes | 2 | |
| 5.b | even | 2 | 1 | 9000.2.a.e | yes | 2 | |
| 15.d | odd | 2 | 1 | 9000.2.a.d | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9000.2.a.d | ✓ | 2 | 15.d | odd | 2 | 1 | |
| 9000.2.a.e | yes | 2 | 5.b | even | 2 | 1 | |
| 9000.2.a.l | yes | 2 | 3.b | odd | 2 | 1 | |
| 9000.2.a.m | yes | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9000))\):
|
\( T_{7}^{2} - 2T_{7} - 4 \)
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\( T_{11}^{2} - 2T_{11} - 4 \)
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\( T_{13}^{2} + 2T_{13} - 4 \)
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\( T_{17}^{2} + 3T_{17} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} \)
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| $7$ |
\( T^{2} - 2T - 4 \)
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| $11$ |
\( T^{2} - 2T - 4 \)
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| $13$ |
\( T^{2} + 2T - 4 \)
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| $17$ |
\( T^{2} + 3T + 1 \)
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| $19$ |
\( T^{2} + 3T + 1 \)
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| $23$ |
\( T^{2} - 3T + 1 \)
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| $29$ |
\( T^{2} + 8T - 4 \)
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| $31$ |
\( T^{2} + 5T - 5 \)
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| $37$ |
\( T^{2} - 8T - 4 \)
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| $41$ |
\( T^{2} - 6T - 36 \)
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| $43$ |
\( (T + 10)^{2} \)
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| $47$ |
\( T^{2} - 9T + 19 \)
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| $53$ |
\( T^{2} + T - 11 \)
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| $59$ |
\( T^{2} + 6T + 4 \)
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| $61$ |
\( T^{2} + 11T + 19 \)
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| $67$ |
\( T^{2} - 6T + 4 \)
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| $71$ |
\( T^{2} + 14T + 44 \)
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| $73$ |
\( T^{2} + 2T - 44 \)
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| $79$ |
\( T^{2} + 11T + 29 \)
|
| $83$ |
\( T^{2} - T - 61 \)
|
| $89$ |
\( T^{2} - 20T + 80 \)
|
| $97$ |
\( T^{2} + 6T - 36 \)
|
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