Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [403,2,Mod(1,403)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(403, 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("403.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 403 = 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 403.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: | \(3.21797120146\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.5748973.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{6} - x^{5} - 6x^{4} + 4x^{3} + 8x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 5\nu^{3} + 3\nu^{2} + 4\nu - 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} - 2\nu^{2} + 6\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 6\beta_{3} + 8\beta_{2} + 12\beta _1 + 2 \)
|
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 |
|
−2.61388 | −1.18785 | 4.83234 | 1.45573 | 3.10489 | −1.87247 | −7.40339 | −1.58901 | −3.80510 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.94648 | 0.310892 | 1.78879 | −3.27007 | −0.605146 | 3.06335 | 0.411120 | −2.90335 | 6.36512 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.917109 | −2.85232 | −1.15891 | −0.732299 | 2.61588 | 4.57774 | 2.89707 | 5.13571 | 0.671598 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.482827 | −0.300210 | −1.76688 | 0.848172 | −0.144950 | −1.74674 | −1.81875 | −2.90987 | 0.409520 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0.543189 | 1.35805 | −1.70495 | −3.27954 | 0.737678 | 0.540055 | −2.01249 | −1.15570 | −1.78141 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.45145 | −2.32857 | 4.00960 | −4.02200 | −5.70836 | −4.56194 | 4.92644 | 2.42222 | −9.85973 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(-1\) |
| \(31\) | \(-1\) |
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 | 403.2.a.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 3627.2.a.m | 6 | ||
| 4.b | odd | 2 | 1 | 6448.2.a.y | 6 | ||
| 13.b | even | 2 | 1 | 5239.2.a.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 403.2.a.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 3627.2.a.m | 6 | 3.b | odd | 2 | 1 | ||
| 5239.2.a.g | 6 | 13.b | even | 2 | 1 | ||
| 6448.2.a.y | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} - 7T_{2}^{4} - 13T_{2}^{3} + 6T_{2}^{2} + 7T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(403))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots - 3 \)
|
| $3$ |
\( T^{6} + 5 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} + 9 T^{5} + \cdots + 39 \)
|
| $7$ |
\( T^{6} - 29 T^{4} + \cdots - 113 \)
|
| $11$ |
\( T^{6} + 5 T^{5} + \cdots + 9 \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} + 23 T^{5} + \cdots - 1821 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots + 2779 \)
|
| $23$ |
\( T^{6} + 18 T^{5} + \cdots + 2271 \)
|
| $29$ |
\( T^{6} + 18 T^{5} + \cdots + 1821 \)
|
| $31$ |
\( (T - 1)^{6} \)
|
| $37$ |
\( T^{6} + 13 T^{5} + \cdots + 2371 \)
|
| $41$ |
\( T^{6} + 5 T^{5} + \cdots - 9633 \)
|
| $43$ |
\( T^{6} + 7 T^{5} + \cdots + 599 \)
|
| $47$ |
\( T^{6} + 9 T^{5} + \cdots - 5361 \)
|
| $53$ |
\( T^{6} + 31 T^{5} + \cdots - 3633 \)
|
| $59$ |
\( T^{6} + T^{5} + \cdots - 1359 \)
|
| $61$ |
\( T^{6} + 15 T^{5} + \cdots - 6223 \)
|
| $67$ |
\( T^{6} + 28 T^{5} + \cdots + 221807 \)
|
| $71$ |
\( T^{6} - T^{5} + \cdots + 627 \)
|
| $73$ |
\( T^{6} + 20 T^{5} + \cdots + 1721 \)
|
| $79$ |
\( T^{6} + 15 T^{5} + \cdots + 15161 \)
|
| $83$ |
\( T^{6} - T^{5} + \cdots - 4077 \)
|
| $89$ |
\( T^{6} + T^{5} + \cdots - 3681 \)
|
| $97$ |
\( T^{6} + 5 T^{5} + \cdots - 1351 \)
|
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