Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8670,2,Mod(1,8670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8670, 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("8670.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8670.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: | \(69.2302985525\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.45769536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 15x^{4} + 72x^{2} - 109 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{6} - 15x^{4} + 72x^{2} - 109 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 10\nu^{2} + 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} + 23\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 10\beta_{2} + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 10\beta_{3} + 27\beta_1 \)
|
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 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −4.91571 | 1.00000 | 1.00000 | 1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −4.31998 | 1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −1.80310 | 1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | −0.196904 | 1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | 2.31998 | 1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | −1.00000 | 1.00000 | 1.00000 | −1.00000 | 2.91571 | 1.00000 | 1.00000 | 1.00000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(17\) | \(1\) |
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 | 8670.2.a.cg | ✓ | 6 |
| 17.b | even | 2 | 1 | 8670.2.a.ch | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8670.2.a.cg | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8670.2.a.ch | yes | 6 | 17.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8670))\):
|
\( T_{7}^{6} + 6T_{7}^{5} - 12T_{7}^{4} - 88T_{7}^{3} + 39T_{7}^{2} + 270T_{7} + 51 \)
|
|
\( T_{11}^{6} + 6T_{11}^{5} - 18T_{11}^{4} - 170T_{11}^{3} - 171T_{11}^{2} + 612T_{11} + 963 \)
|
|
\( T_{13}^{6} + 6T_{13}^{5} - 30T_{13}^{4} - 160T_{13}^{3} + 141T_{13}^{2} + 618T_{13} - 629 \)
|
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\( T_{23}^{6} + 6T_{23}^{5} - 54T_{23}^{4} - 350T_{23}^{3} - 603T_{23}^{2} - 306T_{23} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T - 1)^{6} \)
|
| $7$ |
\( T^{6} + 6 T^{5} + \cdots + 51 \)
|
| $11$ |
\( T^{6} + 6 T^{5} + \cdots + 963 \)
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| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 629 \)
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| $17$ |
\( T^{6} \)
|
| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 2269 \)
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| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 9 \)
|
| $29$ |
\( T^{6} + 12 T^{5} + \cdots + 24696 \)
|
| $31$ |
\( T^{6} + 30 T^{5} + \cdots + 136 \)
|
| $37$ |
\( T^{6} + 6 T^{5} + \cdots + 1347 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots - 9081 \)
|
| $43$ |
\( T^{6} - 6 T^{5} + \cdots + 28232 \)
|
| $47$ |
\( T^{6} + 18 T^{5} + \cdots - 8991 \)
|
| $53$ |
\( T^{6} + 18 T^{5} + \cdots - 243 \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots - 4293 \)
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| $61$ |
\( T^{6} + 24 T^{5} + \cdots - 141976 \)
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| $67$ |
\( T^{6} - 240 T^{4} + \cdots - 102952 \)
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| $71$ |
\( T^{6} + 24 T^{5} + \cdots + 72 \)
|
| $73$ |
\( T^{6} - 18 T^{5} + \cdots - 856 \)
|
| $79$ |
\( T^{6} + 30 T^{5} + \cdots + 5048 \)
|
| $83$ |
\( T^{6} + 18 T^{5} + \cdots + 1761336 \)
|
| $89$ |
\( T^{6} - 12 T^{5} + \cdots + 3231 \)
|
| $97$ |
\( T^{6} + 18 T^{5} + \cdots - 435608 \)
|
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