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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{24})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 4x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} + 5\beta_{2} ) / 2 \)
|
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 | −2.41421 | 1.00000 | 1.00000 | −2.41421 | −1.00000 | 1.00000 | 2.82843 | 1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.41421 | 1.00000 | 1.00000 | −2.41421 | −1.00000 | 1.00000 | 2.82843 | 1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0.414214 | 1.00000 | 1.00000 | 0.414214 | −1.00000 | 1.00000 | −2.82843 | 1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.414214 | 1.00000 | 1.00000 | 0.414214 | −1.00000 | 1.00000 | −2.82843 | 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.cr | yes | 4 |
| 11.b | odd | 2 | 1 | 8470.2.a.cp | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8470.2.a.cp | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 8470.2.a.cr | yes | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):
|
\( T_{3}^{2} + 2T_{3} - 1 \)
|
|
\( T_{13}^{4} - 22T_{13}^{2} + 48T_{13} - 23 \)
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\( T_{17}^{2} - 4T_{17} - 2 \)
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\( T_{19}^{4} + 12T_{19}^{3} + 26T_{19}^{2} - 60T_{19} - 167 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
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| $3$ |
\( (T^{2} + 2 T - 1)^{2} \)
|
| $5$ |
\( (T - 1)^{4} \)
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| $7$ |
\( (T + 1)^{4} \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 22 T^{2} + \cdots - 23 \)
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| $17$ |
\( (T^{2} - 4 T - 2)^{2} \)
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| $19$ |
\( T^{4} + 12 T^{3} + \cdots - 167 \)
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| $23$ |
\( T^{4} + 8 T^{3} + \cdots + 1 \)
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| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $31$ |
\( T^{4} - 124T^{2} + 2116 \)
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| $37$ |
\( T^{4} + 16 T^{3} + \cdots - 800 \)
|
| $41$ |
\( T^{4} - 8 T^{3} + \cdots - 2396 \)
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| $43$ |
\( T^{4} + 8 T^{3} + \cdots - 284 \)
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| $47$ |
\( T^{4} + 16 T^{3} + \cdots + 772 \)
|
| $53$ |
\( T^{4} - 76 T^{2} + \cdots + 4 \)
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| $59$ |
\( T^{4} + 8 T^{3} + \cdots + 73 \)
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| $61$ |
\( T^{4} + 8 T^{3} + \cdots - 128 \)
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| $67$ |
\( T^{4} - 132T^{2} + 900 \)
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| $71$ |
\( T^{4} + 8 T^{3} + \cdots - 3008 \)
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| $73$ |
\( (T^{2} - 4 T + 2)^{2} \)
|
| $79$ |
\( T^{4} + 24 T^{3} + \cdots - 1031 \)
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| $83$ |
\( T^{4} - 8 T^{3} + \cdots - 5807 \)
|
| $89$ |
\( T^{4} - 148T^{2} + 676 \)
|
| $97$ |
\( T^{4} + 8 T^{3} + \cdots - 2672 \)
|
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