Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6027,2,Mod(1,6027)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, 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("6027.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6027 = 3 \cdot 7^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6027.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: | \(48.1258372982\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.981328.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 10x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 861) |
| 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
\( x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 10x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 6\nu + 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 10\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{4} + 3\beta_{3} + 7\beta_{2} + 9\beta _1 + 16 \)
|
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 |
|
−2.74401 | 1.00000 | 5.52961 | 0.811635 | −2.74401 | 0 | −9.68531 | 1.00000 | −2.22714 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.84433 | 1.00000 | 1.40154 | 1.91836 | −1.84433 | 0 | 1.10376 | 1.00000 | −3.53808 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.24342 | 1.00000 | −0.453913 | −3.27559 | −1.24342 | 0 | 3.05124 | 1.00000 | 4.07293 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.909252 | 1.00000 | −1.17326 | 2.40229 | 0.909252 | 0 | −2.88529 | 1.00000 | 2.18428 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.92250 | 1.00000 | 1.69602 | −2.85669 | 1.92250 | 0 | −0.584397 | 1.00000 | −5.49199 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(41\) | \(1\) |
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 | 6027.2.a.v | 5 | |
| 7.b | odd | 2 | 1 | 861.2.a.j | ✓ | 5 | |
| 21.c | even | 2 | 1 | 2583.2.a.s | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 861.2.a.j | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 2583.2.a.s | 5 | 21.c | even | 2 | 1 | ||
| 6027.2.a.v | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):
|
\( T_{2}^{5} + 3T_{2}^{4} - 4T_{2}^{3} - 14T_{2}^{2} + T_{2} + 11 \)
|
|
\( T_{5}^{5} + T_{5}^{4} - 14T_{5}^{3} - 2T_{5}^{2} + 53T_{5} - 35 \)
|
|
\( T_{13}^{5} + 3T_{13}^{4} - 18T_{13}^{3} - 26T_{13}^{2} + 91T_{13} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 3 T^{4} + \cdots + 11 \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} + T^{4} + \cdots - 35 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 4 T^{4} + \cdots - 80 \)
|
| $13$ |
\( T^{5} + 3 T^{4} + \cdots - 1 \)
|
| $17$ |
\( T^{5} - 8 T^{4} + \cdots + 124 \)
|
| $19$ |
\( T^{5} + 20 T^{4} + \cdots - 1804 \)
|
| $23$ |
\( T^{5} + 5 T^{4} + \cdots + 301 \)
|
| $29$ |
\( T^{5} - 9 T^{4} + \cdots + 5 \)
|
| $31$ |
\( T^{5} + 16 T^{4} + \cdots + 2452 \)
|
| $37$ |
\( T^{5} + 19 T^{4} + \cdots + 779 \)
|
| $41$ |
\( (T + 1)^{5} \)
|
| $43$ |
\( T^{5} - 6 T^{4} + \cdots + 13244 \)
|
| $47$ |
\( T^{5} + 9 T^{4} + \cdots - 3173 \)
|
| $53$ |
\( T^{5} + 29 T^{4} + \cdots - 65531 \)
|
| $59$ |
\( T^{5} + 28 T^{4} + \cdots - 28964 \)
|
| $61$ |
\( T^{5} + 16 T^{4} + \cdots + 7732 \)
|
| $67$ |
\( T^{5} - 21 T^{4} + \cdots - 55 \)
|
| $71$ |
\( T^{5} - 6 T^{4} + \cdots + 1124 \)
|
| $73$ |
\( T^{5} - 4 T^{4} + \cdots - 532 \)
|
| $79$ |
\( T^{5} - 21 T^{4} + \cdots - 887 \)
|
| $83$ |
\( T^{5} + 26 T^{4} + \cdots + 10784 \)
|
| $89$ |
\( T^{5} + 12 T^{4} + \cdots - 2576 \)
|
| $97$ |
\( T^{5} + 37 T^{4} + \cdots - 12545 \)
|
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