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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1197392.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 14x - 2 \)
|
| 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
\( x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 14x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 4\nu^{2} + 7\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{3} - \nu^{2} + 11\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 3\beta_{2} + 10\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -3\beta_{4} + 4\beta_{3} + 13\beta_{2} + 31\beta _1 + 19 \)
|
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 |
|
−2.48439 | 1.00000 | 4.17219 | 4.10940 | −2.48439 | 0 | −5.39657 | 1.00000 | −10.2093 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.612990 | 1.00000 | −1.62424 | 1.48910 | −0.612990 | 0 | 2.22162 | 1.00000 | −0.912805 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.863619 | 1.00000 | −1.25416 | 3.66459 | 0.863619 | 0 | −2.81036 | 1.00000 | 3.16481 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.54585 | 1.00000 | 4.48134 | 2.36161 | 2.54585 | 0 | 6.31711 | 1.00000 | 6.01230 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.68791 | 1.00000 | 5.22488 | −2.62470 | 2.68791 | 0 | 8.66818 | 1.00000 | −7.05495 | ||||||||||||||||||||||||||||||||||
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.x | 5 | |
| 7.b | odd | 2 | 1 | 861.2.a.k | ✓ | 5 | |
| 21.c | even | 2 | 1 | 2583.2.a.q | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 861.2.a.k | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 2583.2.a.q | 5 | 21.c | even | 2 | 1 | ||
| 6027.2.a.x | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):
|
\( T_{2}^{5} - 3T_{2}^{4} - 6T_{2}^{3} + 20T_{2}^{2} - T_{2} - 9 \)
|
|
\( T_{5}^{5} - 9T_{5}^{4} + 18T_{5}^{3} + 42T_{5}^{2} - 171T_{5} + 139 \)
|
|
\( T_{13}^{5} - 3T_{13}^{4} - 56T_{13}^{3} + 192T_{13}^{2} + 473T_{13} - 1681 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 9 \)
|
| $3$ |
\( (T - 1)^{5} \)
|
| $5$ |
\( T^{5} - 9 T^{4} + \cdots + 139 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 24 T^{3} + \cdots + 112 \)
|
| $13$ |
\( T^{5} - 3 T^{4} + \cdots - 1681 \)
|
| $17$ |
\( T^{5} - 16 T^{4} + \cdots + 716 \)
|
| $19$ |
\( T^{5} + 4 T^{4} + \cdots + 68 \)
|
| $23$ |
\( T^{5} + 3 T^{4} + \cdots + 2021 \)
|
| $29$ |
\( T^{5} - 13 T^{4} + \cdots - 269 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots + 860 \)
|
| $37$ |
\( T^{5} - 17 T^{4} + \cdots + 8027 \)
|
| $41$ |
\( (T - 1)^{5} \)
|
| $43$ |
\( T^{5} - 6 T^{4} + \cdots - 4 \)
|
| $47$ |
\( T^{5} - 15 T^{4} + \cdots + 3451 \)
|
| $53$ |
\( T^{5} - 11 T^{4} + \cdots - 9197 \)
|
| $59$ |
\( T^{5} + 12 T^{4} + \cdots - 1884 \)
|
| $61$ |
\( T^{5} + 12 T^{4} + \cdots + 120204 \)
|
| $67$ |
\( T^{5} - 11 T^{4} + \cdots - 64297 \)
|
| $71$ |
\( T^{5} - 18 T^{4} + \cdots + 972 \)
|
| $73$ |
\( T^{5} + 12 T^{4} + \cdots - 2956 \)
|
| $79$ |
\( T^{5} - 23 T^{4} + \cdots - 905 \)
|
| $83$ |
\( T^{5} - 2 T^{4} + \cdots - 8608 \)
|
| $89$ |
\( T^{5} - 28 T^{4} + \cdots - 48 \)
|
| $97$ |
\( T^{5} + 3 T^{4} + \cdots + 263 \)
|
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