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: | 6.6.745749504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 18x^{4} - 4x^{3} + 81x^{2} + 36x - 44 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 18x^{4} - 4x^{3} + 81x^{2} + 36x - 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 9\nu - 2 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 13\nu^{2} - 2\nu + 24 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 15\nu^{3} + 11\nu^{2} + 48\nu - 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{2} + 9\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 13\beta_{3} + 2\beta _1 + 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 4\beta_{4} + 2\beta_{3} + 60\beta_{2} + 89\beta _1 + 30 \)
|
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 | −3.40870 | 1.00000 | −1.00000 | −3.40870 | −1.00000 | 1.00000 | 8.61924 | −1.00000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.67544 | 1.00000 | −1.00000 | −2.67544 | −1.00000 | 1.00000 | 4.15798 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.567932 | 1.00000 | −1.00000 | −0.567932 | −1.00000 | 1.00000 | −2.67745 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.16996 | 1.00000 | −1.00000 | 1.16996 | −1.00000 | 1.00000 | −1.63119 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.23874 | 1.00000 | −1.00000 | 2.23874 | −1.00000 | 1.00000 | 2.01195 | −1.00000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 3.24337 | 1.00000 | −1.00000 | 3.24337 | −1.00000 | 1.00000 | 7.51947 | −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.dd | yes | 6 |
| 11.b | odd | 2 | 1 | 8470.2.a.cx | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8470.2.a.cx | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 8470.2.a.dd | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):
|
\( T_{3}^{6} - 18T_{3}^{4} + 4T_{3}^{3} + 81T_{3}^{2} - 36T_{3} - 44 \)
|
|
\( T_{13}^{6} - 42T_{13}^{4} + 8T_{13}^{3} + 441T_{13}^{2} - 168T_{13} - 956 \)
|
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\( T_{17}^{6} + 6T_{17}^{5} - 51T_{17}^{4} - 352T_{17}^{3} - 48T_{17}^{2} + 1416T_{17} - 716 \)
|
|
\( T_{19}^{6} - 45T_{19}^{4} + 104T_{19}^{3} + 99T_{19}^{2} - 216T_{19} - 143 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 18 T^{4} + \cdots - 44 \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} - 42 T^{4} + \cdots - 956 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots - 716 \)
|
| $19$ |
\( T^{6} - 45 T^{4} + \cdots - 143 \)
|
| $23$ |
\( T^{6} - 54 T^{4} + \cdots + 1188 \)
|
| $29$ |
\( T^{6} + 12 T^{5} + \cdots + 1600 \)
|
| $31$ |
\( T^{6} - 108 T^{4} + \cdots + 13312 \)
|
| $37$ |
\( T^{6} - 24 T^{5} + \cdots + 1024 \)
|
| $41$ |
\( T^{6} + 12 T^{5} + \cdots - 34848 \)
|
| $43$ |
\( T^{6} - 18 T^{5} + \cdots - 1052 \)
|
| $47$ |
\( T^{6} - 24 T^{5} + \cdots + 15696 \)
|
| $53$ |
\( T^{6} - 36 T^{5} + \cdots + 62452 \)
|
| $59$ |
\( T^{6} - 30 T^{5} + \cdots + 126949 \)
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| $61$ |
\( T^{6} + 36 T^{5} + \cdots - 198272 \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots - 22412 \)
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| $71$ |
\( T^{6} - 6 T^{5} + \cdots - 3008 \)
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| $73$ |
\( T^{6} - 6 T^{5} + \cdots - 28908 \)
|
| $79$ |
\( T^{6} - 24 T^{5} + \cdots + 12469 \)
|
| $83$ |
\( T^{6} + 24 T^{5} + \cdots - 132896 \)
|
| $89$ |
\( T^{6} - 36 T^{5} + \cdots + 216928 \)
|
| $97$ |
\( T^{6} - 387 T^{4} + \cdots + 34192 \)
|
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