Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5635,2,Mod(1,5635)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, 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("5635.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5635 = 5 \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5635.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: | \(44.9957015390\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.169449536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 8x^{4} - 2x^{3} + 13x^{2} + 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{6} - 8x^{4} - 2x^{3} + 13x^{2} + 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + \nu^{4} + 7\nu^{3} - 4\nu^{2} - 9\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} - 2\nu^{2} + 13\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{3} + 2\beta_{2} + 27\beta _1 + 11 \)
|
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 |
|
−2.19211 | −2.19211 | 2.80536 | −1.00000 | 4.80536 | 0 | −1.76543 | 1.80536 | 2.19211 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.49211 | −1.49211 | 0.226383 | −1.00000 | 2.22638 | 0 | 2.64643 | −0.773617 | 1.49211 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.497659 | −0.497659 | −1.75234 | −1.00000 | 0.247665 | 0 | 1.86739 | −2.75234 | 0.497659 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.165121 | 0.165121 | −1.97273 | −1.00000 | 0.0272650 | 0 | −0.655983 | −2.97273 | −0.165121 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.44884 | 1.44884 | 0.0991459 | −1.00000 | 2.09915 | 0 | −2.75404 | −0.900854 | −1.44884 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.56791 | 2.56791 | 4.59418 | −1.00000 | 6.59418 | 0 | 6.66164 | 3.59418 | −2.56791 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 5635.2.a.z | ✓ | 6 |
| 7.b | odd | 2 | 1 | 5635.2.a.ba | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5635.2.a.z | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5635.2.a.ba | yes | 6 | 7.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5635))\):
|
\( T_{2}^{6} - 8T_{2}^{4} - 2T_{2}^{3} + 13T_{2}^{2} + 4T_{2} - 1 \)
|
|
\( T_{3}^{6} - 8T_{3}^{4} - 2T_{3}^{3} + 13T_{3}^{2} + 4T_{3} - 1 \)
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\( T_{11}^{6} - 4T_{11}^{5} - 15T_{11}^{4} + 34T_{11}^{3} + 62T_{11}^{2} - 46T_{11} - 27 \)
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\( T_{17}^{6} - 27T_{17}^{4} - 16T_{17}^{3} + 172T_{17}^{2} + 232T_{17} + 79 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 8 T^{4} + \cdots - 1 \)
|
| $3$ |
\( T^{6} - 8 T^{4} + \cdots - 1 \)
|
| $5$ |
\( (T + 1)^{6} \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} - 4 T^{5} + \cdots - 27 \)
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| $13$ |
\( T^{6} - 8 T^{5} + \cdots - 49 \)
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| $17$ |
\( T^{6} - 27 T^{4} + \cdots + 79 \)
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| $19$ |
\( T^{6} + 2 T^{5} + \cdots - 763 \)
|
| $23$ |
\( (T + 1)^{6} \)
|
| $29$ |
\( T^{6} + 8 T^{5} + \cdots - 6976 \)
|
| $31$ |
\( T^{6} - 8 T^{5} + \cdots - 16607 \)
|
| $37$ |
\( T^{6} + 16 T^{5} + \cdots - 20665 \)
|
| $41$ |
\( T^{6} - 22 T^{5} + \cdots - 36739 \)
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| $43$ |
\( T^{6} - 8 T^{5} + \cdots - 19893 \)
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| $47$ |
\( T^{6} + 2 T^{5} + \cdots - 41661 \)
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| $53$ |
\( T^{6} - 4 T^{5} + \cdots - 185 \)
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| $59$ |
\( T^{6} - 18 T^{5} + \cdots + 2121 \)
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| $61$ |
\( T^{6} + 6 T^{5} + \cdots + 5 \)
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| $67$ |
\( T^{6} - 4 T^{5} + \cdots - 8353 \)
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| $71$ |
\( T^{6} - 186 T^{4} + \cdots - 3467 \)
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| $73$ |
\( T^{6} - 12 T^{5} + \cdots + 214587 \)
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| $79$ |
\( T^{6} + 30 T^{5} + \cdots + 91161 \)
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| $83$ |
\( T^{6} - 32 T^{5} + \cdots - 24753 \)
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| $89$ |
\( T^{6} + 28 T^{5} + \cdots - 5747 \)
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| $97$ |
\( T^{6} - 4 T^{5} + \cdots - 382285 \)
|
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