Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8470,2,Mod(1,8470)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, 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("8470.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8470.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: | \(67.6332905120\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.10784448.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 11x^{4} - 4x^{3} + 31x^{2} + 22x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 11x^{4} - 4x^{3} + 31x^{2} + 22x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 6\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} - \nu^{3} + 7\nu^{2} + 6\nu - 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 5\nu^{2} - 8\nu - 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 7\nu^{2} - 2\nu - 8 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - \nu^{4} - 17\nu^{3} + 3\nu^{2} + 32\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 + 8 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{4} - 3\beta_{2} - 2\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{4} + 14\beta_{3} + 3\beta_{2} - 9\beta _1 + 44 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{5} + 37\beta_{4} + 4\beta_{3} - 35\beta_{2} - 21\beta _1 + 38 ) / 2 \)
|
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 |
|
−1.00000 | −1.34292 | 1.00000 | 1.00000 | 1.34292 | −1.00000 | −1.00000 | −1.19656 | −1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.34292 | 1.00000 | 1.00000 | 1.34292 | −1.00000 | −1.00000 | −1.19656 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.529317 | 1.00000 | 1.00000 | −0.529317 | −1.00000 | −1.00000 | −2.71982 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.529317 | 1.00000 | 1.00000 | −0.529317 | −1.00000 | −1.00000 | −2.71982 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.81361 | 1.00000 | 1.00000 | −2.81361 | −1.00000 | −1.00000 | 4.91638 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.81361 | 1.00000 | 1.00000 | −2.81361 | −1.00000 | −1.00000 | 4.91638 | −1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(1\) |
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 | 8470.2.a.cz | ✓ | 6 |
| 11.b | odd | 2 | 1 | 8470.2.a.df | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8470.2.a.cz | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8470.2.a.df | yes | 6 | 11.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):
|
\( T_{3}^{3} - 2T_{3}^{2} - 3T_{3} + 2 \)
|
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\( T_{13}^{6} - 42T_{13}^{4} + 441T_{13}^{2} - 108 \)
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\( T_{17}^{3} + T_{17}^{2} - 24T_{17} + 38 \)
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\( T_{19}^{6} - 41T_{19}^{4} - 80T_{19}^{3} + 283T_{19}^{2} + 944T_{19} + 709 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
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| $3$ |
\( (T^{3} - 2 T^{2} - 3 T + 2)^{2} \)
|
| $5$ |
\( (T - 1)^{6} \)
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| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} - 42 T^{4} + \cdots - 108 \)
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| $17$ |
\( (T^{3} + T^{2} - 24 T + 38)^{2} \)
|
| $19$ |
\( T^{6} - 41 T^{4} + \cdots + 709 \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots + 2932 \)
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| $29$ |
\( T^{6} + 8 T^{5} + \cdots + 256 \)
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| $31$ |
\( (T^{3} - 2 T^{2} - 6 T + 8)^{2} \)
|
| $37$ |
\( T^{6} - 4 T^{5} + \cdots - 512 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots - 32 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 1012 \)
|
| $47$ |
\( T^{6} - 16 T^{5} + \cdots - 26864 \)
|
| $53$ |
\( T^{6} - 12 T^{5} + \cdots + 436 \)
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| $59$ |
\( T^{6} - 22 T^{5} + \cdots - 19679 \)
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| $61$ |
\( T^{6} - 4 T^{5} + \cdots - 5888 \)
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| $67$ |
\( T^{6} - 20 T^{5} + \cdots + 45748 \)
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| $71$ |
\( T^{6} - 14 T^{5} + \cdots - 402176 \)
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| $73$ |
\( T^{6} + 18 T^{5} + \cdots - 3788 \)
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| $79$ |
\( T^{6} + 32 T^{5} + \cdots - 197027 \)
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| $83$ |
\( T^{6} - 16 T^{5} + \cdots - 25136 \)
|
| $89$ |
\( T^{6} - 4 T^{5} + \cdots + 529504 \)
|
| $97$ |
\( T^{6} - 4 T^{5} + \cdots + 46912 \)
|
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