Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4000,2,Mod(1,4000)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4000, 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("4000.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4000 = 2^{5} \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4000.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: | \(31.9401608085\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.30040000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 6\nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 11\nu^{3} - 2\nu^{2} + 29\nu + 10 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 11\nu^{3} + 4\nu^{2} - 31\nu - 18 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{5} + 2\nu^{4} + 31\nu^{3} - 6\nu^{2} - 79\nu - 26 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 6\beta _1 + 2 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 6\beta_{4} + 9\beta_{3} + \beta_{2} + 8\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 4\beta_{3} + 11\beta_{2} + 39\beta _1 + 20 \)
|
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 | −2.71210 | 0 | 0 | 0 | −0.0156607 | 0 | 4.35547 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.29226 | 0 | 0 | 0 | 0.938845 | 0 | 2.25447 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.306558 | 0 | 0 | 0 | 2.60656 | 0 | −2.90602 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.481965 | 0 | 0 | 0 | −2.69841 | 0 | −2.76771 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.81030 | 0 | 0 | 0 | 4.37760 | 0 | 0.277175 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.40554 | 0 | 0 | 0 | −2.20893 | 0 | 2.78662 | 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 | 4000.2.a.n | yes | 6 |
| 4.b | odd | 2 | 1 | 4000.2.a.k | ✓ | 6 | |
| 5.b | even | 2 | 1 | 4000.2.a.l | yes | 6 | |
| 5.c | odd | 4 | 2 | 4000.2.c.g | 12 | ||
| 8.b | even | 2 | 1 | 8000.2.a.bw | 6 | ||
| 8.d | odd | 2 | 1 | 8000.2.a.bv | 6 | ||
| 20.d | odd | 2 | 1 | 4000.2.a.m | yes | 6 | |
| 20.e | even | 4 | 2 | 4000.2.c.f | 12 | ||
| 40.e | odd | 2 | 1 | 8000.2.a.bx | 6 | ||
| 40.f | even | 2 | 1 | 8000.2.a.bu | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4000.2.a.k | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 4000.2.a.l | yes | 6 | 5.b | even | 2 | 1 | |
| 4000.2.a.m | yes | 6 | 20.d | odd | 2 | 1 | |
| 4000.2.a.n | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 4000.2.c.f | 12 | 20.e | even | 4 | 2 | ||
| 4000.2.c.g | 12 | 5.c | odd | 4 | 2 | ||
| 8000.2.a.bu | 6 | 40.f | even | 2 | 1 | ||
| 8000.2.a.bv | 6 | 8.d | odd | 2 | 1 | ||
| 8000.2.a.bw | 6 | 8.b | even | 2 | 1 | ||
| 8000.2.a.bx | 6 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4000))\):
|
\( T_{3}^{6} - 11T_{3}^{4} + 4T_{3}^{3} + 29T_{3}^{2} - 22T_{3} + 4 \)
|
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\( T_{7}^{6} - 3T_{7}^{5} - 15T_{7}^{4} + 30T_{7}^{3} + 55T_{7}^{2} - 63T_{7} - 1 \)
|
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\( T_{11}^{6} - 13T_{11}^{5} + 41T_{11}^{4} + 50T_{11}^{3} - 325T_{11}^{2} + 375T_{11} - 125 \)
|
|
\( T_{13}^{6} - 7T_{13}^{5} - 34T_{13}^{4} + 245T_{13}^{3} + 150T_{13}^{2} - 1375T_{13} + 625 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 11 T^{4} + \cdots + 4 \)
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| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots - 1 \)
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| $11$ |
\( T^{6} - 13 T^{5} + \cdots - 125 \)
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| $13$ |
\( T^{6} - 7 T^{5} + \cdots + 625 \)
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| $17$ |
\( T^{6} + 4 T^{5} + \cdots + 2000 \)
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| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 125 \)
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| $23$ |
\( T^{6} - 7 T^{5} + \cdots - 64 \)
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| $29$ |
\( T^{6} - 2 T^{5} + \cdots - 2636 \)
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| $31$ |
\( T^{6} - 12 T^{5} + \cdots - 29500 \)
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| $37$ |
\( T^{6} - 21 T^{5} + \cdots - 8000 \)
|
| $41$ |
\( T^{6} + 5 T^{5} + \cdots + 10475 \)
|
| $43$ |
\( T^{6} + 8 T^{5} + \cdots - 24256 \)
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| $47$ |
\( T^{6} - 19 T^{5} + \cdots + 2549 \)
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| $53$ |
\( T^{6} - 11 T^{5} + \cdots - 9875 \)
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| $59$ |
\( T^{6} - 25 T^{5} + \cdots - 59375 \)
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| $61$ |
\( T^{6} + 6 T^{5} + \cdots - 32220 \)
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| $67$ |
\( T^{6} - 159 T^{4} + \cdots - 5364 \)
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| $71$ |
\( T^{6} - 34 T^{5} + \cdots - 372500 \)
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| $73$ |
\( T^{6} + 20 T^{5} + \cdots + 2500 \)
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| $79$ |
\( T^{6} - 16 T^{5} + \cdots + 174500 \)
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| $83$ |
\( T^{6} + 4 T^{5} + \cdots - 46080 \)
|
| $89$ |
\( T^{6} + 3 T^{5} + \cdots - 142144 \)
|
| $97$ |
\( T^{6} + 20 T^{5} + \cdots + 177500 \)
|
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