Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [501,2,Mod(1,501)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(501, 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("501.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 501 = 3 \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 501.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: | \(4.00050514127\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 8x^{6} + 15x^{5} + 19x^{4} - 31x^{3} - 13x^{2} + 14x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 7\nu^{5} + 6\nu^{4} + 13\nu^{3} - 8\nu^{2} - 5\nu - 1 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 9\nu^{5} - 8\nu^{4} + 23\nu^{3} + 16\nu^{2} - 13\nu - 3 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( -\nu^{7} + \nu^{6} + 8\nu^{5} - 7\nu^{4} - 18\nu^{3} + 13\nu^{2} + 9\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 5\beta_{2} + 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 2\beta_{5} + \beta_{4} + 5\beta_{3} + 16\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + 8\beta_{4} + 23\beta_{2} + 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} + \beta_{6} + 17\beta_{5} + 9\beta_{4} + 22\beta_{3} + \beta_{2} + 65\beta _1 + 18 \)
|
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 |
|
−2.79278 | −1.00000 | 5.79962 | −0.621044 | 2.79278 | −0.794861 | −10.6115 | 1.00000 | 1.73444 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.16366 | −1.00000 | 2.68143 | −3.62511 | 2.16366 | −1.70865 | −1.47437 | 1.00000 | 7.84350 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.08371 | −1.00000 | 2.34184 | 1.56175 | 2.08371 | 4.60397 | −0.712305 | 1.00000 | −3.25424 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.850180 | −1.00000 | −1.27719 | 4.14293 | 0.850180 | −1.03675 | 2.78621 | 1.00000 | −3.52224 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.271170 | −1.00000 | −1.92647 | −2.71255 | 0.271170 | −5.14356 | 1.06474 | 1.00000 | 0.735563 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.431433 | −1.00000 | −1.81387 | 0.925650 | −0.431433 | 2.76498 | −1.64543 | 1.00000 | 0.399356 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.30229 | −1.00000 | 3.30055 | 1.28584 | −2.30229 | −0.120874 | 2.99424 | 1.00000 | 2.96038 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.42777 | −1.00000 | 3.89409 | 0.0425261 | −2.42777 | 1.43576 | 4.59842 | 1.00000 | 0.103244 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(167\) | \(-1\) |
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 | 501.2.a.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1503.2.a.f | 8 | ||
| 4.b | odd | 2 | 1 | 8016.2.a.z | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 501.2.a.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1503.2.a.f | 8 | 3.b | odd | 2 | 1 | ||
| 8016.2.a.z | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 3T_{2}^{7} - 10T_{2}^{6} - 34T_{2}^{5} + 17T_{2}^{4} + 100T_{2}^{3} + 43T_{2}^{2} - 21T_{2} - 7 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 3 T^{7} + \cdots - 7 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} - T^{7} - 21 T^{6} + \cdots - 2 \)
|
| $7$ |
\( T^{8} - 31 T^{6} + \cdots - 16 \)
|
| $11$ |
\( T^{8} - 5 T^{7} + \cdots + 500 \)
|
| $13$ |
\( T^{8} - 61 T^{6} + \cdots + 9224 \)
|
| $17$ |
\( T^{8} + 7 T^{7} + \cdots - 1058 \)
|
| $19$ |
\( T^{8} - 24 T^{7} + \cdots - 19872 \)
|
| $23$ |
\( T^{8} - T^{7} + \cdots - 202112 \)
|
| $29$ |
\( T^{8} + 11 T^{7} + \cdots + 5992 \)
|
| $31$ |
\( T^{8} - 30 T^{7} + \cdots - 115552 \)
|
| $37$ |
\( T^{8} - 11 T^{7} + \cdots - 464216 \)
|
| $41$ |
\( T^{8} - 10 T^{7} + \cdots + 389482 \)
|
| $43$ |
\( T^{8} - 24 T^{7} + \cdots + 322 \)
|
| $47$ |
\( T^{8} + 3 T^{7} + \cdots + 355556 \)
|
| $53$ |
\( T^{8} + 25 T^{7} + \cdots + 191494 \)
|
| $59$ |
\( T^{8} - 45 T^{7} + \cdots + 288064 \)
|
| $61$ |
\( T^{8} - 16 T^{7} + \cdots - 64516 \)
|
| $67$ |
\( T^{8} - 18 T^{7} + \cdots - 209558 \)
|
| $71$ |
\( T^{8} - 21 T^{7} + \cdots - 12220528 \)
|
| $73$ |
\( T^{8} + 8 T^{7} + \cdots - 14504 \)
|
| $79$ |
\( T^{8} - 10 T^{7} + \cdots + 2662174 \)
|
| $83$ |
\( T^{8} - 7 T^{7} + \cdots - 55504 \)
|
| $89$ |
\( T^{8} - 26 T^{7} + \cdots + 99128 \)
|
| $97$ |
\( T^{8} + 3 T^{7} + \cdots + 4584952 \)
|
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