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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.36538000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 13x^{4} - 4x^{3} + 41x^{2} + 16x - 16 \)
|
| 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
\( x^{6} - 13x^{4} - 4x^{3} + 41x^{2} + 16x - 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} + 3\nu^{3} + 24\nu^{2} - 55\nu - 44 ) / 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} + 7\nu^{3} - 24\nu^{2} - 15\nu + 4 ) / 10 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{5} + 12\nu^{4} + 31\nu^{3} - 112\nu^{2} - 75\nu + 172 ) / 40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{5} - 2\nu^{4} - 31\nu^{3} + 2\nu^{2} + 65\nu + 18 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 7\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - 7\beta_{4} + 11\beta_{3} + 11\beta_{2} + 12\beta _1 + 36 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} - 4\beta_{4} + 17\beta_{3} + 48\beta_{2} + 58\beta _1 + 56 \)
|
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 | −2.52231 | 1.00000 | 1.42123 | 2.52231 | 1.00000 | −1.00000 | 3.36207 | −1.42123 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.14489 | 1.00000 | 2.45305 | 2.14489 | 1.00000 | −1.00000 | 1.60057 | −2.45305 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.969024 | 1.00000 | −3.11111 | 0.969024 | 1.00000 | −1.00000 | −2.06099 | 3.11111 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.479869 | 1.00000 | −2.85512 | −0.479869 | 1.00000 | −1.00000 | −2.76973 | 2.85512 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.04245 | 1.00000 | 2.05192 | −2.04245 | 1.00000 | −1.00000 | 1.17159 | −2.05192 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 3.11392 | 1.00000 | −0.959972 | −3.11392 | 1.00000 | −1.00000 | 6.69649 | 0.959972 | |||||||||||||||||||||||||||||||||||||
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.z | ✓ | 6 |
| 19.b | odd | 2 | 1 | 5054.2.a.be | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 5054.2.a.z | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 5054.2.a.be | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):
|
\( T_{3}^{6} - 13T_{3}^{4} - 4T_{3}^{3} + 41T_{3}^{2} + 16T_{3} - 16 \)
|
|
\( T_{5}^{6} + T_{5}^{5} - 15T_{5}^{4} - 6T_{5}^{3} + 67T_{5}^{2} - 7T_{5} - 61 \)
|
|
\( T_{13}^{6} + 15T_{13}^{5} + 70T_{13}^{4} + 73T_{13}^{3} - 250T_{13}^{2} - 525T_{13} - 109 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{6} \)
|
| $3$ |
\( T^{6} - 13 T^{4} + \cdots - 16 \)
|
| $5$ |
\( T^{6} + T^{5} + \cdots - 61 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 4 T^{5} + \cdots + 16 \)
|
| $13$ |
\( T^{6} + 15 T^{5} + \cdots - 109 \)
|
| $17$ |
\( T^{6} + 9 T^{5} + \cdots + 4975 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - 4 T^{5} + \cdots - 16 \)
|
| $29$ |
\( T^{6} + 7 T^{5} + \cdots - 12625 \)
|
| $31$ |
\( T^{6} + 4 T^{5} + \cdots - 14384 \)
|
| $37$ |
\( T^{6} + 3 T^{5} + \cdots - 41 \)
|
| $41$ |
\( T^{6} + 11 T^{5} + \cdots - 11449 \)
|
| $43$ |
\( T^{6} + 10 T^{5} + \cdots + 2224 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 11056 \)
|
| $53$ |
\( T^{6} - 5 T^{5} + \cdots + 2111 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 59120 \)
|
| $61$ |
\( T^{6} + 21 T^{5} + \cdots + 1625711 \)
|
| $67$ |
\( T^{6} + 14 T^{5} + \cdots - 390224 \)
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| $71$ |
\( T^{6} + 24 T^{5} + \cdots + 256 \)
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| $73$ |
\( T^{6} + 21 T^{5} + \cdots - 57301 \)
|
| $79$ |
\( T^{6} + 58 T^{5} + \cdots - 407920 \)
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| $83$ |
\( T^{6} - 20 T^{5} + \cdots + 649984 \)
|
| $89$ |
\( T^{6} + 7 T^{5} + \cdots + 150355 \)
|
| $97$ |
\( T^{6} + 7 T^{5} + \cdots + 3254519 \)
|
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