Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [786,4,Mod(1,786)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(786, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("786.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 786 = 2 \cdot 3 \cdot 131 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 786.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: | \(46.3755012645\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.98582301.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 27x^{3} - 14x^{2} + 81x + 51 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 27x^{3} - 14x^{2} + 81x + 51 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -5\nu^{4} - \nu^{3} + 172\nu^{2} - 63\nu - 678 ) / 93 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{4} + 17\nu^{3} + 238\nu^{2} - 231\nu - 936 ) / 93 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10\nu^{4} + 2\nu^{3} - 251\nu^{2} - 246\nu + 426 ) / 93 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{4} + 16\nu^{3} + 317\nu^{2} - 201\nu - 591 ) / 93 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{4} - \beta_{3} + 4\beta_{2} + 2\beta _1 + 34 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{4} - 3\beta_{3} + 10\beta_{2} - 9\beta _1 + 17 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -122\beta_{4} - 20\beta_{3} + 119\beta_{2} + 31\beta _1 + 740 ) / 3 \)
|
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 |
|
2.00000 | 3.00000 | 4.00000 | −12.0098 | 6.00000 | 6.17416 | 8.00000 | 9.00000 | −24.0195 | |||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | −11.2790 | 6.00000 | 7.58190 | 8.00000 | 9.00000 | −22.5581 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.00000 | 4.00000 | −5.40708 | 6.00000 | −25.8818 | 8.00000 | 9.00000 | −10.8142 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.00000 | 4.00000 | 3.25204 | 6.00000 | 1.44882 | 8.00000 | 9.00000 | 6.50409 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 3.00000 | 4.00000 | 13.4438 | 6.00000 | −33.3231 | 8.00000 | 9.00000 | 26.8877 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(131\) | \(-1\) |
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 | 786.4.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 2358.4.a.c | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 786.4.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 2358.4.a.c | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 12T_{5}^{4} - 174T_{5}^{3} - 2377T_{5}^{2} - 801T_{5} + 32022 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(786))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{5} \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( T^{5} + 12 T^{4} + \cdots + 32022 \)
|
| $7$ |
\( T^{5} + 44 T^{4} + \cdots - 58494 \)
|
| $11$ |
\( T^{5} + 169 T^{4} + \cdots + 12105117 \)
|
| $13$ |
\( T^{5} + 110 T^{4} + \cdots + 5672859 \)
|
| $17$ |
\( T^{5} + \cdots - 1792610973 \)
|
| $19$ |
\( T^{5} + 200 T^{4} + \cdots + 281853293 \)
|
| $23$ |
\( T^{5} + 110 T^{4} + \cdots + 876101184 \)
|
| $29$ |
\( T^{5} + \cdots - 1059456861 \)
|
| $31$ |
\( T^{5} + \cdots - 569894909341 \)
|
| $37$ |
\( T^{5} + \cdots + 29730417848 \)
|
| $41$ |
\( T^{5} + \cdots + 243538985988 \)
|
| $43$ |
\( T^{5} + \cdots + 115137496062 \)
|
| $47$ |
\( T^{5} + \cdots + 716840828928 \)
|
| $53$ |
\( T^{5} + \cdots + 8205069164508 \)
|
| $59$ |
\( T^{5} + \cdots - 791324086101 \)
|
| $61$ |
\( T^{5} + \cdots + 116771787939681 \)
|
| $67$ |
\( T^{5} + \cdots - 26482674516984 \)
|
| $71$ |
\( T^{5} + \cdots - 14488716257464 \)
|
| $73$ |
\( T^{5} + \cdots - 278282161962 \)
|
| $79$ |
\( T^{5} + \cdots - 397928798748568 \)
|
| $83$ |
\( T^{5} + \cdots + 486906730866282 \)
|
| $89$ |
\( T^{5} + \cdots + 533567270807306 \)
|
| $97$ |
\( T^{5} + \cdots + 159895482588912 \)
|
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