Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5312,2,Mod(1,5312)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5312, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5312.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5312 = 2^{6} \cdot 83 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5312.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: | \(42.4165335537\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.9059636.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 83) |
| 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
\( x^{6} - x^{5} - 8x^{4} + 11x^{3} + 4x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{4} + \nu^{3} - 6\nu^{2} - \nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 7\nu^{2} + 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 7\nu^{3} + 12\nu^{2} - 2\nu - 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 8\nu^{3} + 11\nu^{2} + 5\nu - 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( -2\nu^{5} + 2\nu^{4} + 15\nu^{3} - 22\nu^{2} - \nu + 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + \beta_{2} - \beta _1 + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 5\beta_{4} - 9\beta_{3} - 9\beta_{2} + 9\beta _1 - 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} - 2\beta_{4} + 16\beta_{3} + 10\beta_{2} - 9\beta _1 + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -24\beta_{5} + 33\beta_{4} - 79\beta_{3} - 69\beta_{2} + 71\beta _1 - 92 ) / 2 \)
|
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 |
|
0 | −2.04941 | 0 | −2.79494 | 0 | −3.61008 | 0 | 1.20006 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.49589 | 0 | 3.47366 | 0 | 1.39854 | 0 | −0.762314 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.32239 | 0 | −4.05060 | 0 | 2.22837 | 0 | −1.25129 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.13227 | 0 | 1.45742 | 0 | −3.35950 | 0 | −1.71797 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.96807 | 0 | 1.62860 | 0 | −3.12266 | 0 | 0.873293 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.76735 | 0 | −1.71413 | 0 | 3.46533 | 0 | 4.65821 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(83\) | \(-1\) |
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 | 5312.2.a.bo | 6 | |
| 4.b | odd | 2 | 1 | 5312.2.a.bn | 6 | ||
| 8.b | even | 2 | 1 | 1328.2.a.l | 6 | ||
| 8.d | odd | 2 | 1 | 83.2.a.b | ✓ | 6 | |
| 24.f | even | 2 | 1 | 747.2.a.j | 6 | ||
| 40.e | odd | 2 | 1 | 2075.2.a.g | 6 | ||
| 56.e | even | 2 | 1 | 4067.2.a.d | 6 | ||
| 664.h | even | 2 | 1 | 6889.2.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 83.2.a.b | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 747.2.a.j | 6 | 24.f | even | 2 | 1 | ||
| 1328.2.a.l | 6 | 8.b | even | 2 | 1 | ||
| 2075.2.a.g | 6 | 40.e | odd | 2 | 1 | ||
| 4067.2.a.d | 6 | 56.e | even | 2 | 1 | ||
| 5312.2.a.bn | 6 | 4.b | odd | 2 | 1 | ||
| 5312.2.a.bo | 6 | 1.a | even | 1 | 1 | trivial | |
| 6889.2.a.e | 6 | 664.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5312))\):
|
\( T_{3}^{6} - T_{3}^{5} - 10T_{3}^{4} + 5T_{3}^{3} + 30T_{3}^{2} - 4T_{3} - 25 \)
|
|
\( T_{5}^{6} + 2T_{5}^{5} - 20T_{5}^{4} - 28T_{5}^{3} + 104T_{5}^{2} + 64T_{5} - 160 \)
|
|
\( T_{7}^{6} + 3T_{7}^{5} - 22T_{7}^{4} - 55T_{7}^{3} + 154T_{7}^{2} + 228T_{7} - 409 \)
|
|
\( T_{11}^{6} + 3T_{11}^{5} - 26T_{11}^{4} - 83T_{11}^{3} + 66T_{11}^{2} + 156T_{11} - 113 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + \cdots - 25 \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots - 160 \)
|
| $7$ |
\( T^{6} + 3 T^{5} + \cdots - 409 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots - 113 \)
|
| $13$ |
\( T^{6} + 14 T^{5} + \cdots + 992 \)
|
| $17$ |
\( T^{6} + 5 T^{5} + \cdots - 275 \)
|
| $19$ |
\( T^{6} + 4 T^{5} + \cdots + 6176 \)
|
| $23$ |
\( T^{6} - 5 T^{5} + \cdots + 10912 \)
|
| $29$ |
\( T^{6} - T^{5} + \cdots - 55 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots - 313 \)
|
| $37$ |
\( T^{6} + 39 T^{5} + \cdots - 91499 \)
|
| $41$ |
\( T^{6} + T^{5} + \cdots - 248 \)
|
| $43$ |
\( T^{6} + 8 T^{5} + \cdots + 6400 \)
|
| $47$ |
\( T^{6} - 12 T^{5} + \cdots + 25952 \)
|
| $53$ |
\( T^{6} + 14 T^{5} + \cdots - 64 \)
|
| $59$ |
\( T^{6} + 17 T^{5} + \cdots + 3527 \)
|
| $61$ |
\( T^{6} - 5 T^{5} + \cdots - 47347 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 264256 \)
|
| $71$ |
\( T^{6} - 26 T^{5} + \cdots + 7232 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 39136 \)
|
| $79$ |
\( T^{6} - 12 T^{5} + \cdots - 160 \)
|
| $83$ |
\( (T - 1)^{6} \)
|
| $89$ |
\( T^{6} + 22 T^{5} + \cdots + 144896 \)
|
| $97$ |
\( T^{6} - 6 T^{5} + \cdots - 101120 \)
|
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