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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 886x^{3} + 514x^{2} + 139837x + 674496 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 190) |
| 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
\( x^{5} - 2x^{4} - 886x^{3} + 514x^{2} + 139837x + 674496 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -26\nu^{4} + 230\nu^{3} + 19891\nu^{2} - 158335\nu - 1868973 ) / 4083 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13\nu^{4} - 115\nu^{3} - 10626\nu^{2} + 81209\nu + 1174703 ) / 1361 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 37\nu^{4} - 432\nu^{3} - 30662\nu^{2} + 292483\nu + 3547012 ) / 1361 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{3} - 3\beta_{2} + 3\beta _1 + 353 \)
|
| \(\nu^{3}\) | \(=\) |
\( -13\beta_{4} + 45\beta_{3} + 12\beta_{2} + 574\beta _1 + 533 \)
|
| \(\nu^{4}\) | \(=\) |
\( -115\beta_{4} - 1132\beta_{3} - 2346\beta_{2} + 1283\beta _1 + 202890 \)
|
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 | −21.2130 | 16.0000 | 0 | −84.8518 | −87.2771 | 64.0000 | 206.989 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −2.88726 | 16.0000 | 0 | −11.5490 | 245.889 | 64.0000 | −234.664 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −2.22714 | 16.0000 | 0 | −8.90858 | −166.998 | 64.0000 | −238.040 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 22.4322 | 16.0000 | 0 | 89.7288 | 171.699 | 64.0000 | 260.203 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 30.8952 | 16.0000 | 0 | 123.581 | −226.313 | 64.0000 | 711.511 | 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.m | 5 | |
| 5.b | even | 2 | 1 | 190.6.a.h | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 190.6.a.h | ✓ | 5 | 5.b | even | 2 | 1 | |
| 950.6.a.m | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 27T_{3}^{4} - 596T_{3}^{3} + 12254T_{3}^{2} + 72372T_{3} + 94536 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(950))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( T^{5} - 27 T^{4} + \cdots + 94536 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + \cdots + 139260303296 \)
|
| $11$ |
\( T^{5} + \cdots - 1330573116416 \)
|
| $13$ |
\( T^{5} + \cdots + 6515063936328 \)
|
| $17$ |
\( T^{5} + \cdots + 4531307288208 \)
|
| $19$ |
\( (T + 361)^{5} \)
|
| $23$ |
\( T^{5} + \cdots + 21\!\cdots\!12 \)
|
| $29$ |
\( T^{5} + \cdots - 48\!\cdots\!20 \)
|
| $31$ |
\( T^{5} + \cdots - 73\!\cdots\!08 \)
|
| $37$ |
\( T^{5} + \cdots + 85\!\cdots\!64 \)
|
| $41$ |
\( T^{5} + \cdots - 24\!\cdots\!72 \)
|
| $43$ |
\( T^{5} + \cdots - 16\!\cdots\!96 \)
|
| $47$ |
\( T^{5} + \cdots - 49\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 20\!\cdots\!36 \)
|
| $59$ |
\( T^{5} + \cdots - 67\!\cdots\!20 \)
|
| $61$ |
\( T^{5} + \cdots + 11\!\cdots\!32 \)
|
| $67$ |
\( T^{5} + \cdots + 90\!\cdots\!84 \)
|
| $71$ |
\( T^{5} + \cdots + 21\!\cdots\!76 \)
|
| $73$ |
\( T^{5} + \cdots + 80\!\cdots\!36 \)
|
| $79$ |
\( T^{5} + \cdots + 84\!\cdots\!80 \)
|
| $83$ |
\( T^{5} + \cdots + 70\!\cdots\!12 \)
|
| $89$ |
\( T^{5} + \cdots - 84\!\cdots\!00 \)
|
| $97$ |
\( T^{5} + \cdots - 34\!\cdots\!48 \)
|
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