Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5265,2,Mod(1,5265)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, 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("5265.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5265 = 3^{4} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5265.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: | \(42.0412366642\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.585163476.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 13x^{3} + 14x^{2} - 11x - 1 \)
|
| 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
\( x^{6} - 2x^{5} - 8x^{4} + 13x^{3} + 14x^{2} - 11x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu - 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 13\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + 7\beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 8\beta_{3} + 36\beta _1 + 10 \)
|
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 |
|
−2.34309 | 0 | 3.49005 | 1.00000 | 0 | −0.543460 | −3.49133 | 0 | −2.34309 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.23897 | 0 | −0.464949 | 1.00000 | 0 | 4.61495 | 3.05400 | 0 | −1.23897 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.0828741 | 0 | −1.99313 | 1.00000 | 0 | −1.04171 | 0.330927 | 0 | −0.0828741 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.667718 | 0 | −1.55415 | 1.00000 | 0 | 2.14402 | −2.37317 | 0 | 0.667718 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.36425 | 0 | 3.58967 | 1.00000 | 0 | 3.65786 | 3.75837 | 0 | 2.36425 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.63297 | 0 | 4.93251 | 1.00000 | 0 | −4.83166 | 7.72120 | 0 | 2.63297 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(1\) |
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 | 5265.2.a.x | yes | 6 |
| 3.b | odd | 2 | 1 | 5265.2.a.w | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5265.2.a.w | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 5265.2.a.x | yes | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5265))\):
|
\( T_{2}^{6} - 2T_{2}^{5} - 8T_{2}^{4} + 13T_{2}^{3} + 14T_{2}^{2} - 11T_{2} - 1 \)
|
|
\( T_{7}^{6} - 4T_{7}^{5} - 24T_{7}^{4} + 103T_{7}^{3} + 24T_{7}^{2} - 203T_{7} - 99 \)
|
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\( T_{11}^{6} - 5T_{11}^{5} - 14T_{11}^{4} + 67T_{11}^{3} - 22T_{11}^{2} - 62T_{11} + 29 \)
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\( T_{17}^{6} - 7T_{17}^{5} - 8T_{17}^{4} + 113T_{17}^{3} - 106T_{17}^{2} - 220T_{17} + 200 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 2 T^{5} + \cdots - 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( T^{6} - 4 T^{5} + \cdots - 99 \)
|
| $11$ |
\( T^{6} - 5 T^{5} + \cdots + 29 \)
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| $13$ |
\( (T + 1)^{6} \)
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| $17$ |
\( T^{6} - 7 T^{5} + \cdots + 200 \)
|
| $19$ |
\( T^{6} - 12 T^{5} + \cdots + 2643 \)
|
| $23$ |
\( T^{6} - 9 T^{5} + \cdots - 864 \)
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| $29$ |
\( T^{6} - 20 T^{5} + \cdots + 1 \)
|
| $31$ |
\( T^{6} - 14 T^{5} + \cdots + 2592 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots - 14724 \)
|
| $41$ |
\( T^{6} - 16 T^{5} + \cdots + 22992 \)
|
| $43$ |
\( T^{6} - 7 T^{5} + \cdots + 4 \)
|
| $47$ |
\( T^{6} - 20 T^{5} + \cdots + 35692 \)
|
| $53$ |
\( T^{6} - 106 T^{4} + \cdots - 24201 \)
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| $59$ |
\( T^{6} - 5 T^{5} + \cdots + 24057 \)
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| $61$ |
\( T^{6} - 16 T^{5} + \cdots + 15647 \)
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| $67$ |
\( T^{6} + 9 T^{5} + \cdots + 74892 \)
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| $71$ |
\( T^{6} - 18 T^{5} + \cdots + 42027 \)
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| $73$ |
\( T^{6} - T^{5} + \cdots - 372276 \)
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| $79$ |
\( T^{6} - 9 T^{5} + \cdots + 972 \)
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| $83$ |
\( T^{6} + 2 T^{5} + \cdots - 2991 \)
|
| $89$ |
\( T^{6} + 11 T^{5} + \cdots - 324 \)
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| $97$ |
\( T^{6} + 11 T^{5} + \cdots - 253168 \)
|
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