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: | \(4\) |
| Coefficient field: | 4.4.10025.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 11x^{2} + 10x + 20 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1000) |
| 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
\( x^{4} - x^{3} - 11x^{2} + 10x + 20 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 7\nu - 6 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 3\nu^{2} - 7\nu - 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + 3\beta_{2} + 7\beta _1 + 1 \)
|
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 | −1.30060 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0 | 0 | 1.13558 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 0 | 0 | 4.48246 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 0 | 0 | 4.68257 | 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.bb | 4 | |
| 3.b | odd | 2 | 1 | 1000.2.a.g | yes | 4 | |
| 5.b | even | 2 | 1 | 9000.2.a.q | 4 | ||
| 12.b | even | 2 | 1 | 2000.2.a.n | 4 | ||
| 15.d | odd | 2 | 1 | 1000.2.a.f | ✓ | 4 | |
| 15.e | even | 4 | 2 | 1000.2.c.c | 8 | ||
| 24.f | even | 2 | 1 | 8000.2.a.bo | 4 | ||
| 24.h | odd | 2 | 1 | 8000.2.a.bd | 4 | ||
| 60.h | even | 2 | 1 | 2000.2.a.q | 4 | ||
| 60.l | odd | 4 | 2 | 2000.2.c.i | 8 | ||
| 120.i | odd | 2 | 1 | 8000.2.a.bn | 4 | ||
| 120.m | even | 2 | 1 | 8000.2.a.be | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1000.2.a.f | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 1000.2.a.g | yes | 4 | 3.b | odd | 2 | 1 | |
| 1000.2.c.c | 8 | 15.e | even | 4 | 2 | ||
| 2000.2.a.n | 4 | 12.b | even | 2 | 1 | ||
| 2000.2.a.q | 4 | 60.h | even | 2 | 1 | ||
| 2000.2.c.i | 8 | 60.l | odd | 4 | 2 | ||
| 8000.2.a.bd | 4 | 24.h | odd | 2 | 1 | ||
| 8000.2.a.be | 4 | 120.m | even | 2 | 1 | ||
| 8000.2.a.bn | 4 | 120.i | odd | 2 | 1 | ||
| 8000.2.a.bo | 4 | 24.f | even | 2 | 1 | ||
| 9000.2.a.q | 4 | 5.b | even | 2 | 1 | ||
| 9000.2.a.bb | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9000))\):
|
\( T_{7}^{4} - 9T_{7}^{3} + 18T_{7}^{2} + 17T_{7} - 31 \)
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\( T_{11}^{4} + 5T_{11}^{3} - 2T_{11}^{2} - 25T_{11} + 1 \)
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\( T_{13}^{4} + 4T_{13}^{3} - 17T_{13}^{2} - 2T_{13} + 19 \)
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\( T_{17}^{4} - 4T_{17}^{3} - 21T_{17}^{2} + 100T_{17} - 80 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 9 T^{3} + \cdots - 31 \)
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| $11$ |
\( T^{4} + 5 T^{3} + \cdots + 1 \)
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| $13$ |
\( T^{4} + 4 T^{3} + \cdots + 19 \)
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| $17$ |
\( T^{4} - 4 T^{3} + \cdots - 80 \)
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| $19$ |
\( T^{4} - 57 T^{2} + \cdots + 191 \)
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| $23$ |
\( T^{4} + 11 T^{3} + \cdots - 316 \)
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| $29$ |
\( T^{4} + 10 T^{3} + \cdots - 4 \)
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| $31$ |
\( T^{4} - 9 T^{3} + \cdots - 1780 \)
|
| $37$ |
\( T^{4} + 15 T^{3} + \cdots - 404 \)
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| $41$ |
\( T^{4} + 17 T^{3} + \cdots + 139 \)
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| $43$ |
\( T^{4} + 12 T^{3} + \cdots - 16 \)
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| $47$ |
\( (T^{2} + 7 T - 19)^{2} \)
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| $53$ |
\( T^{4} + 6 T^{3} + \cdots - 709 \)
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| $59$ |
\( T^{4} + 18 T^{3} + \cdots - 61 \)
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| $61$ |
\( T^{4} - 11 T^{3} + \cdots - 100 \)
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| $67$ |
\( T^{4} - T^{3} + \cdots + 8524 \)
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| $71$ |
\( T^{4} + 26 T^{3} + \cdots - 20 \)
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| $73$ |
\( T^{4} + 9 T^{3} + \cdots + 2524 \)
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| $79$ |
\( T^{4} + 8 T^{3} + \cdots + 2420 \)
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| $83$ |
\( T^{4} - 12 T^{3} + \cdots - 5696 \)
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| $89$ |
\( T^{4} + 5 T^{3} + \cdots - 1084 \)
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| $97$ |
\( T^{4} + 29 T^{3} + \cdots - 8804 \)
|
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