Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5054,2,Mod(1,5054)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5054, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("5054.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5054 = 2 \cdot 7 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5054.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: | \(40.3563931816\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.48952000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 13x^{4} + 44x^{2} + 36x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{6} - x^{5} - 13x^{4} + 44x^{2} + 36x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 10\nu^{3} + 9\nu^{2} + 28\nu + 6 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 9\nu^{3} + 20\nu^{2} + 22\nu - 12 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 9\nu^{3} - 16\nu^{2} - 30\nu - 4 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 15\nu^{3} - 6\nu^{2} - 54\nu - 16 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 2\beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + \beta_{3} + 2\beta_{2} + 9\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 13\beta_{4} + 8\beta_{3} + 6\beta_{2} + 25\beta _1 + 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{5} + 37\beta_{4} + 17\beta_{3} + 36\beta_{2} + 94\beta _1 + 82 \)
|
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 | −3.16418 | 1.00000 | −2.51032 | 3.16418 | −1.00000 | −1.00000 | 7.01204 | 2.51032 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.81341 | 1.00000 | 0.208819 | 1.81341 | −1.00000 | −1.00000 | 0.288464 | −0.208819 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0.295818 | 1.00000 | 1.81581 | −0.295818 | −1.00000 | −1.00000 | −2.91249 | −1.81581 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 1.63229 | 1.00000 | −2.92353 | −1.63229 | −1.00000 | −1.00000 | −0.335615 | 2.92353 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 1.75047 | 1.00000 | 2.63573 | −1.75047 | −1.00000 | −1.00000 | 0.0641566 | −2.63573 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 3.29901 | 1.00000 | −4.22651 | −3.29901 | −1.00000 | −1.00000 | 7.88345 | 4.22651 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(1\) |
| \(19\) | \(-1\) |
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 | 5054.2.a.ba | ✓ | 6 |
| 19.b | odd | 2 | 1 | 5054.2.a.bd | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5054.2.a.ba | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5054.2.a.bd | yes | 6 | 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 13T_{3}^{4} + 26T_{3}^{3} + 27T_{3}^{2} - 64T_{3} + 16 \)
|
|
\( T_{5}^{6} + 5T_{5}^{5} - 9T_{5}^{4} - 56T_{5}^{3} + 19T_{5}^{2} + 147T_{5} - 31 \)
|
|
\( T_{13}^{6} - 23T_{13}^{5} + 194T_{13}^{4} - 661T_{13}^{3} + 212T_{13}^{2} + 3579T_{13} - 5571 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 16 \)
|
| $5$ |
\( T^{6} + 5 T^{5} + \cdots - 31 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots + 656 \)
|
| $13$ |
\( T^{6} - 23 T^{5} + \cdots - 5571 \)
|
| $17$ |
\( T^{6} + 3 T^{5} + \cdots - 3799 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + 4 T^{5} + \cdots + 944 \)
|
| $29$ |
\( T^{6} - 5 T^{5} + \cdots + 19 \)
|
| $31$ |
\( T^{6} - 12 T^{5} + \cdots + 80816 \)
|
| $37$ |
\( T^{6} - T^{5} + \cdots + 271 \)
|
| $41$ |
\( T^{6} - 19 T^{5} + \cdots + 17209 \)
|
| $43$ |
\( T^{6} + 2 T^{5} + \cdots - 656 \)
|
| $47$ |
\( T^{6} + 12 T^{5} + \cdots + 8656 \)
|
| $53$ |
\( T^{6} + 11 T^{5} + \cdots + 128959 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 38096 \)
|
| $61$ |
\( T^{6} + T^{5} + \cdots - 171 \)
|
| $67$ |
\( T^{6} + 14 T^{5} + \cdots - 23824 \)
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| $71$ |
\( T^{6} + 8 T^{5} + \cdots - 80896 \)
|
| $73$ |
\( T^{6} - 5 T^{5} + \cdots - 35951 \)
|
| $79$ |
\( T^{6} - 30 T^{5} + \cdots - 31984 \)
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| $83$ |
\( T^{6} + 12 T^{5} + \cdots + 2304 \)
|
| $89$ |
\( T^{6} - 11 T^{5} + \cdots - 392551 \)
|
| $97$ |
\( T^{6} - 25 T^{5} + \cdots + 2987609 \)
|
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