Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8010,2,Mod(1,8010)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8010, 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("8010.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8010.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: | \(63.9601720190\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.21712324.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 15x^{3} + 27x^{2} + 14x - 29 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2670) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 15x^{3} + 27x^{2} + 14x - 29 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 16\nu^{2} + 11\nu + 21 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{4} - 3\nu^{3} - 80\nu^{2} + 25\nu + 111 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( 3\nu^{4} - 2\nu^{3} - 47\nu^{2} + 17\nu + 59 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 5\beta_{2} + 15\beta _1 - 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{4} - 15\beta_{3} - 19\beta_{2} + 20\beta _1 + 88 \)
|
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 | 0 | 1.00000 | 1.00000 | 0 | −4.51319 | −1.00000 | 0 | −1.00000 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | −1.76451 | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | 1.96432 | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | 2.97977 | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | 1.00000 | 0 | 4.33360 | −1.00000 | 0 | −1.00000 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(89\) | \(-1\) |
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 | 8010.2.a.bh | 5 | |
| 3.b | odd | 2 | 1 | 2670.2.a.r | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2670.2.a.r | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 8010.2.a.bh | 5 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8010))\):
|
\( T_{7}^{5} - 3T_{7}^{4} - 23T_{7}^{3} + 72T_{7}^{2} + 58T_{7} - 202 \)
|
|
\( T_{11}^{5} + T_{11}^{4} - 45T_{11}^{3} + 18T_{11}^{2} + 540T_{11} - 892 \)
|
|
\( T_{13}^{5} - 34T_{13}^{3} - 10T_{13}^{2} + 268T_{13} + 88 \)
|
|
\( T_{17}^{5} - 3T_{17}^{4} - 13T_{17}^{3} + 20T_{17}^{2} + 20T_{17} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 1)^{5} \)
|
| $7$ |
\( T^{5} - 3 T^{4} + \cdots - 202 \)
|
| $11$ |
\( T^{5} + T^{4} + \cdots - 892 \)
|
| $13$ |
\( T^{5} - 34 T^{3} + \cdots + 88 \)
|
| $17$ |
\( T^{5} - 3 T^{4} + \cdots + 4 \)
|
| $19$ |
\( T^{5} - 9 T^{4} + \cdots - 1556 \)
|
| $23$ |
\( T^{5} + 6 T^{4} + \cdots - 352 \)
|
| $29$ |
\( T^{5} + 6 T^{4} + \cdots - 352 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots + 664 \)
|
| $37$ |
\( T^{5} - 10 T^{4} + \cdots + 296 \)
|
| $41$ |
\( T^{5} + T^{4} + \cdots + 136 \)
|
| $43$ |
\( T^{5} - 14 T^{4} + \cdots - 128 \)
|
| $47$ |
\( T^{5} - 7 T^{4} + \cdots - 8104 \)
|
| $53$ |
\( T^{5} + 20 T^{4} + \cdots + 3328 \)
|
| $59$ |
\( T^{5} + 8 T^{4} + \cdots + 832 \)
|
| $61$ |
\( T^{5} - 23 T^{4} + \cdots + 3994 \)
|
| $67$ |
\( T^{5} - 15 T^{4} + \cdots - 115636 \)
|
| $71$ |
\( T^{5} + 24 T^{4} + \cdots - 3424 \)
|
| $73$ |
\( T^{5} - 6 T^{4} + \cdots - 50864 \)
|
| $79$ |
\( T^{5} - 27 T^{4} + \cdots - 488 \)
|
| $83$ |
\( T^{5} + 9 T^{4} + \cdots + 20752 \)
|
| $89$ |
\( (T - 1)^{5} \)
|
| $97$ |
\( T^{5} + 10 T^{4} + \cdots - 544 \)
|
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