Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [503,1,Mod(502,503)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(503, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("503.502");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 503 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 503.b (of order \(2\), degree \(1\), minimal) |
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: | \(0.251029701354\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{21})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 6x^{3} + 8x^{2} - 8x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{21}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{21} - \cdots)\) |
$q$-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 \(\nu = \zeta_{21} + \zeta_{21}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 5\nu^{3} + \nu^{2} + 5\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 5\beta_{3} - \beta_{2} + 10\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/503\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 502.1 |
|
−1.80194 | 0.149460 | 2.24698 | 0 | −0.269318 | 1.91115 | −2.24698 | −0.977662 | 0 | ||||||||||||||||||||||||||||||||||||
| 502.2 | −1.80194 | 1.65248 | 2.24698 | 0 | −2.97766 | −1.46610 | −2.24698 | 1.73068 | 0 | |||||||||||||||||||||||||||||||||||||
| 502.3 | −0.445042 | −1.46610 | −0.801938 | 0 | 0.652478 | −1.97766 | 0.801938 | 1.14946 | 0 | |||||||||||||||||||||||||||||||||||||
| 502.4 | −0.445042 | 1.91115 | −0.801938 | 0 | −0.850540 | 0.730682 | 0.801938 | 2.65248 | 0 | |||||||||||||||||||||||||||||||||||||
| 502.5 | 1.24698 | −1.97766 | 0.554958 | 0 | −2.46610 | 1.65248 | −0.554958 | 2.91115 | 0 | |||||||||||||||||||||||||||||||||||||
| 502.6 | 1.24698 | 0.730682 | 0.554958 | 0 | 0.911146 | 0.149460 | −0.554958 | −0.466104 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 503.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-503}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 503.1.b.c | ✓ | 6 |
| 503.b | odd | 2 | 1 | CM | 503.1.b.c | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 503.1.b.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 503.1.b.c | ✓ | 6 | 503.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(503, [\chi])\):
|
\( T_{2}^{3} + T_{2}^{2} - 2T_{2} - 1 \)
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\( T_{3}^{6} - T_{3}^{5} - 6T_{3}^{4} + 6T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
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| $3$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $11$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $13$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $17$ |
\( T^{6} \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( (T + 1)^{6} \)
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| $29$ |
\( T^{6} \)
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| $31$ |
\( T^{6} \)
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| $37$ |
\( T^{6} \)
|
| $41$ |
\( T^{6} \)
|
| $43$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $47$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $53$ |
\( T^{6} \)
|
| $59$ |
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
|
| $61$ |
\( (T + 1)^{6} \)
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| $67$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $71$ |
\( T^{6} \)
|
| $73$ |
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
|
| $79$ |
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
|
| $83$ |
\( T^{6} - T^{5} - 6 T^{4} + \cdots + 1 \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( (T^{3} + T^{2} - 2 T - 1)^{2} \)
|
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