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: | \(4\) |
| Coefficient field: | 4.4.4400.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 7x^{2} + 11 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 770) |
| 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 a root \(\nu\) of
\( x^{4} - 7x^{2} + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
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 | −3.14896 | 1.00000 | 1.00000 | 3.14896 | −1.00000 | −1.00000 | 6.91596 | −1.00000 | ||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.54336 | 1.00000 | 1.00000 | 2.54336 | −1.00000 | −1.00000 | 3.46869 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.543362 | 1.00000 | 1.00000 | −0.543362 | −1.00000 | −1.00000 | −2.70476 | −1.00000 | |||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.14896 | 1.00000 | 1.00000 | −1.14896 | −1.00000 | −1.00000 | −1.67989 | −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.co | 4 | |
| 11.b | odd | 2 | 1 | 8470.2.a.cs | 4 | ||
| 11.c | even | 5 | 2 | 770.2.n.f | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.n.f | ✓ | 8 | 11.c | even | 5 | 2 | |
| 8470.2.a.co | 4 | 1.a | even | 1 | 1 | trivial | |
| 8470.2.a.cs | 4 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8470))\):
|
\( T_{3}^{4} + 4T_{3}^{3} - T_{3}^{2} - 10T_{3} + 5 \)
|
|
\( T_{13}^{4} - 14T_{13}^{3} + 54T_{13}^{2} - 220 \)
|
|
\( T_{17}^{4} - 12T_{17}^{3} + 41T_{17}^{2} - 30T_{17} - 25 \)
|
|
\( T_{19}^{4} + 2T_{19}^{3} - 29T_{19}^{2} - 10T_{19} + 145 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{4} \)
|
| $3$ |
\( T^{4} + 4 T^{3} + \cdots + 5 \)
|
| $5$ |
\( (T - 1)^{4} \)
|
| $7$ |
\( (T + 1)^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} - 14 T^{3} + \cdots - 220 \)
|
| $17$ |
\( T^{4} - 12 T^{3} + \cdots - 25 \)
|
| $19$ |
\( T^{4} + 2 T^{3} + \cdots + 145 \)
|
| $23$ |
\( T^{4} - 32T^{2} + 176 \)
|
| $29$ |
\( T^{4} - 10 T^{3} + \cdots - 164 \)
|
| $31$ |
\( T^{4} + 18 T^{3} + \cdots - 4684 \)
|
| $37$ |
\( T^{4} - 18 T^{3} + \cdots - 1100 \)
|
| $41$ |
\( T^{4} - 24 T^{3} + \cdots + 839 \)
|
| $43$ |
\( T^{4} - 10 T^{3} + \cdots + 971 \)
|
| $47$ |
\( T^{4} + 8 T^{3} + \cdots - 1616 \)
|
| $53$ |
\( T^{4} - 20 T^{3} + \cdots - 2804 \)
|
| $59$ |
\( T^{4} + 14 T^{3} + \cdots - 155 \)
|
| $61$ |
\( T^{4} - 14 T^{3} + \cdots - 284 \)
|
| $67$ |
\( T^{4} - 23 T^{2} + \cdots - 29 \)
|
| $71$ |
\( T^{4} + 14 T^{3} + \cdots - 556 \)
|
| $73$ |
\( T^{4} - 30 T^{3} + \cdots - 26249 \)
|
| $79$ |
\( T^{4} - 8 T^{3} + \cdots + 1420 \)
|
| $83$ |
\( T^{4} - 8 T^{3} + \cdots - 1055 \)
|
| $89$ |
\( T^{4} + 4 T^{3} + \cdots - 2749 \)
|
| $97$ |
\( T^{4} + 10 T^{3} + \cdots - 1429 \)
|
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