Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10000,2,Mod(1,10000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("10000.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10000 = 2^{4} \cdot 5^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10000.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: | \(79.8504020213\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.7625.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 9x^{2} + 4x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 50) |
| 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
\( x^{4} - x^{3} - 9x^{2} + 4x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 5\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \)
|
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 | −2.33275 | 0 | 0 | 0 | 3.77447 | 0 | 2.44172 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.34841 | 0 | 0 | 0 | −0.833366 | 0 | −1.18178 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.71472 | 0 | 0 | 0 | −2.77447 | 0 | −0.0597522 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 2.96645 | 0 | 0 | 0 | 1.83337 | 0 | 5.79981 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
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 | 10000.2.a.x | 4 | |
| 4.b | odd | 2 | 1 | 1250.2.a.f | 4 | ||
| 5.b | even | 2 | 1 | 10000.2.a.t | 4 | ||
| 20.d | odd | 2 | 1 | 1250.2.a.l | 4 | ||
| 20.e | even | 4 | 2 | 1250.2.b.e | 8 | ||
| 25.e | even | 10 | 2 | 400.2.u.d | 8 | ||
| 100.h | odd | 10 | 2 | 50.2.d.b | ✓ | 8 | |
| 100.j | odd | 10 | 2 | 250.2.d.d | 8 | ||
| 100.l | even | 20 | 4 | 250.2.e.c | 16 | ||
| 300.r | even | 10 | 2 | 450.2.h.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.2.d.b | ✓ | 8 | 100.h | odd | 10 | 2 | |
| 250.2.d.d | 8 | 100.j | odd | 10 | 2 | ||
| 250.2.e.c | 16 | 100.l | even | 20 | 4 | ||
| 400.2.u.d | 8 | 25.e | even | 10 | 2 | ||
| 450.2.h.e | 8 | 300.r | even | 10 | 2 | ||
| 1250.2.a.f | 4 | 4.b | odd | 2 | 1 | ||
| 1250.2.a.l | 4 | 20.d | odd | 2 | 1 | ||
| 1250.2.b.e | 8 | 20.e | even | 4 | 2 | ||
| 10000.2.a.t | 4 | 5.b | even | 2 | 1 | ||
| 10000.2.a.x | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(10000))\):
|
\( T_{3}^{4} - T_{3}^{3} - 9T_{3}^{2} + 4T_{3} + 16 \)
|
|
\( T_{7}^{4} - 2T_{7}^{3} - 11T_{7}^{2} + 12T_{7} + 16 \)
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\( T_{11}^{4} - 2T_{11}^{3} - 11T_{11}^{2} + 12T_{11} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - T^{3} + \cdots + 16 \)
|
| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
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| $11$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
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| $13$ |
\( T^{4} + 11 T^{3} + \cdots - 199 \)
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| $17$ |
\( T^{4} + 12 T^{3} + \cdots - 109 \)
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| $19$ |
\( T^{4} - 5 T^{3} + \cdots + 80 \)
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| $23$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
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| $29$ |
\( T^{4} - 15 T^{3} + \cdots + 5 \)
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| $31$ |
\( T^{4} - 12 T^{3} + \cdots - 1264 \)
|
| $37$ |
\( T^{4} + 12 T^{3} + \cdots + 71 \)
|
| $41$ |
\( T^{4} - 13 T^{3} + \cdots - 89 \)
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| $43$ |
\( T^{4} - 6 T^{3} + \cdots + 176 \)
|
| $47$ |
\( T^{4} - 2 T^{3} + \cdots + 16 \)
|
| $53$ |
\( T^{4} + 11 T^{3} + \cdots + 256 \)
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| $59$ |
\( T^{4} - 140 T^{2} + \cdots - 320 \)
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| $61$ |
\( T^{4} - 8 T^{3} + \cdots - 1709 \)
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| $67$ |
\( T^{4} - 22 T^{3} + \cdots - 944 \)
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| $71$ |
\( T^{4} - 22 T^{3} + \cdots - 64 \)
|
| $73$ |
\( T^{4} + 21 T^{3} + \cdots - 1084 \)
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| $79$ |
\( T^{4} - 10 T^{3} + \cdots - 320 \)
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| $83$ |
\( T^{4} + 24 T^{3} + \cdots - 3664 \)
|
| $89$ |
\( T^{4} + 5 T^{3} + \cdots - 3100 \)
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| $97$ |
\( (T^{2} + 11 T - 1)^{2} \)
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