Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5415,2,Mod(1,5415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5415, 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("5415.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5415 = 3 \cdot 5 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5415.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: | \(43.2389926945\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.2591125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 6x^{3} + 10x^{2} - 9x + 1 \)
|
| 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
\( x^{6} - x^{5} - 7x^{4} + 6x^{3} + 10x^{2} - 9x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 8\nu^{3} - 4\nu^{2} + 11\nu - 2 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 8\nu^{3} - 7\nu^{2} + 11\nu + 4 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + \nu^{4} + 13\nu^{3} - 4\nu^{2} - 16\nu + 4 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + \nu^{4} + 16\nu^{3} - 7\nu^{2} - 28\nu + 13 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 4\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 5\beta_{3} + 7\beta_{2} + 2\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{5} - 8\beta_{4} - 7\beta_{3} + 8\beta_{2} + 19\beta _1 - 5 \)
|
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 |
|
−2.41183 | −1.00000 | 3.81693 | 1.00000 | 2.41183 | −0.761140 | −4.38212 | 1.00000 | −2.41183 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.36923 | −1.00000 | −0.125197 | 1.00000 | 1.36923 | −3.87497 | 2.90989 | 1.00000 | −1.36923 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.718012 | −1.00000 | −1.48446 | 1.00000 | 0.718012 | 1.91079 | 2.50188 | 1.00000 | −0.718012 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.131653 | −1.00000 | −1.98267 | 1.00000 | 0.131653 | 4.22798 | 0.524331 | 1.00000 | −0.131653 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.51181 | −1.00000 | 0.285565 | 1.00000 | −1.51181 | 2.08642 | −2.59190 | 1.00000 | 1.51181 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.11892 | −1.00000 | 2.48983 | 1.00000 | −2.11892 | −1.58908 | 1.03791 | 1.00000 | 2.11892 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 5415.2.a.bd | ✓ | 6 |
| 19.b | odd | 2 | 1 | 5415.2.a.bg | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5415.2.a.bd | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5415.2.a.bg | yes | 6 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5415))\):
|
\( T_{2}^{6} + T_{2}^{5} - 7T_{2}^{4} - 6T_{2}^{3} + 10T_{2}^{2} + 9T_{2} + 1 \)
|
|
\( T_{7}^{6} - 2T_{7}^{5} - 20T_{7}^{4} + 33T_{7}^{3} + 72T_{7}^{2} - 76T_{7} - 79 \)
|
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\( T_{11}^{6} - 6T_{11}^{5} + 5T_{11}^{4} + 13T_{11}^{3} - 4T_{11}^{2} - 7T_{11} - 1 \)
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\( T_{13}^{6} + 4T_{13}^{5} - 31T_{13}^{4} - 96T_{13}^{3} + 323T_{13}^{2} + 540T_{13} - 1061 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} - 7 T^{4} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} - 2 T^{5} + \cdots - 79 \)
|
| $11$ |
\( T^{6} - 6 T^{5} + \cdots - 1 \)
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| $13$ |
\( T^{6} + 4 T^{5} + \cdots - 1061 \)
|
| $17$ |
\( T^{6} - T^{5} + \cdots + 625 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 3 T^{5} + \cdots + 89 \)
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| $29$ |
\( T^{6} + 27 T^{5} + \cdots + 78355 \)
|
| $31$ |
\( T^{6} + 17 T^{5} + \cdots - 1321 \)
|
| $37$ |
\( T^{6} + 17 T^{5} + \cdots - 711 \)
|
| $41$ |
\( T^{6} + 11 T^{5} + \cdots + 3559 \)
|
| $43$ |
\( T^{6} - 4 T^{5} + \cdots + 72001 \)
|
| $47$ |
\( T^{6} + 14 T^{5} + \cdots + 5011 \)
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| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 179 \)
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| $59$ |
\( T^{6} + 19 T^{5} + \cdots - 29545 \)
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| $61$ |
\( T^{6} + 16 T^{5} + \cdots - 6649 \)
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| $67$ |
\( T^{6} + 3 T^{5} + \cdots - 673781 \)
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| $71$ |
\( T^{6} + 3 T^{5} + \cdots - 1106279 \)
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| $73$ |
\( T^{6} - 8 T^{5} + \cdots + 39349 \)
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| $79$ |
\( T^{6} + 19 T^{5} + \cdots + 7145 \)
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| $83$ |
\( T^{6} + T^{5} + \cdots - 2841471 \)
|
| $89$ |
\( T^{6} + 31 T^{5} + \cdots - 201375 \)
|
| $97$ |
\( T^{6} + 7 T^{5} + \cdots - 303601 \)
|
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