Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,6,Mod(1,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 950.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: | \(152.364628822\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 738x^{4} + 2678x^{3} + 138431x^{2} - 304764x - 7805817 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 738x^{4} + 2678x^{3} + 138431x^{2} - 304764x - 7805817 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3\nu - 248 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{5} + 202\nu^{4} - 2525\nu^{3} - 73951\nu^{2} + 54072\nu + 4765698 ) / 12753 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{5} + 101\nu^{4} - 1971\nu^{3} - 47603\nu^{2} + 246671\nu + 4504098 ) / 4251 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -35\nu^{5} + 179\nu^{4} + 21143\nu^{3} - 173300\nu^{2} - 1953969\nu + 15186861 ) / 12753 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3\beta _1 + 248 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{4} + 9\beta_{3} - 15\beta_{2} + 355\beta _1 - 726 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{5} + 57\beta_{4} - 33\beta_{3} + 610\beta_{2} - 3159\beta _1 + 88928 \)
|
| \(\nu^{5}\) | \(=\) |
\( -303\beta_{5} - 3333\beta_{4} + 5268\beta_{3} - 10893\beta_{2} + 157321\beta _1 - 777807 \)
|
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 |
|
4.00000 | −20.1521 | 16.0000 | 0 | −80.6083 | −92.8279 | 64.0000 | 163.106 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −18.1490 | 16.0000 | 0 | −72.5959 | 218.063 | 64.0000 | 86.3852 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −13.3547 | 16.0000 | 0 | −53.4186 | −177.436 | 64.0000 | −64.6533 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 6.98743 | 16.0000 | 0 | 27.9497 | −78.9467 | 64.0000 | −194.176 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 9.04656 | 16.0000 | 0 | 36.1862 | 114.914 | 64.0000 | −161.160 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | 21.6217 | 16.0000 | 0 | 86.4868 | −37.7667 | 64.0000 | 224.498 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(19\) | \(1\) |
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 | 950.6.a.p | yes | 6 |
| 5.b | even | 2 | 1 | 950.6.a.n | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 950.6.a.n | ✓ | 6 | 5.b | even | 2 | 1 | |
| 950.6.a.p | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 14T_{3}^{5} - 658T_{3}^{4} - 8342T_{3}^{3} + 105051T_{3}^{2} + 803088T_{3} - 6675669 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{6} \)
|
| $3$ |
\( T^{6} + 14 T^{5} + \cdots - 6675669 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 1230596407731 \)
|
| $11$ |
\( T^{6} + \cdots - 105744834074025 \)
|
| $13$ |
\( T^{6} + \cdots + 893703766098549 \)
|
| $17$ |
\( T^{6} + \cdots + 22\!\cdots\!41 \)
|
| $19$ |
\( (T + 361)^{6} \)
|
| $23$ |
\( T^{6} + \cdots - 93\!\cdots\!65 \)
|
| $29$ |
\( T^{6} + \cdots + 74\!\cdots\!25 \)
|
| $31$ |
\( T^{6} + \cdots + 35\!\cdots\!55 \)
|
| $37$ |
\( T^{6} + \cdots + 13\!\cdots\!17 \)
|
| $41$ |
\( T^{6} + \cdots + 12\!\cdots\!29 \)
|
| $43$ |
\( T^{6} + \cdots + 36\!\cdots\!11 \)
|
| $47$ |
\( T^{6} + \cdots - 46\!\cdots\!25 \)
|
| $53$ |
\( T^{6} + \cdots - 42\!\cdots\!59 \)
|
| $59$ |
\( T^{6} + \cdots - 53\!\cdots\!40 \)
|
| $61$ |
\( T^{6} + \cdots + 69\!\cdots\!73 \)
|
| $67$ |
\( T^{6} + \cdots + 55\!\cdots\!27 \)
|
| $71$ |
\( T^{6} + \cdots - 60\!\cdots\!25 \)
|
| $73$ |
\( T^{6} + \cdots + 90\!\cdots\!49 \)
|
| $79$ |
\( T^{6} + \cdots - 52\!\cdots\!25 \)
|
| $83$ |
\( T^{6} + \cdots - 89\!\cdots\!01 \)
|
| $89$ |
\( T^{6} + \cdots - 74\!\cdots\!75 \)
|
| $97$ |
\( T^{6} + \cdots - 86\!\cdots\!19 \)
|
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