Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,6,Mod(1,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, 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("650.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.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: | \(104.249482878\) |
| 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: |
\( x^{5} - 2x^{4} - 809x^{3} + 2922x^{2} + 77904x - 57024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3\cdot 5 \) |
| 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
\( x^{5} - 2x^{4} - 809x^{3} + 2922x^{2} + 77904x - 57024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 10\nu^{3} - 529\nu^{2} + 8554\nu - 37128 ) / 300 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} - 10\nu^{3} + 689\nu^{2} + 4266\nu - 35712 ) / 180 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 10\nu^{3} + 829\nu^{2} - 7654\nu - 60672 ) / 300 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{2} - 3\beta _1 + 326 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{4} - 9\beta_{3} - 7\beta_{2} + 617\beta _1 - 1034 \)
|
| \(\nu^{4}\) | \(=\) |
\( 609\beta_{4} - 90\beta_{3} + 759\beta_{2} - 3971\beta _1 + 199242 \)
|
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 | −24.8846 | 16.0000 | 0 | −99.5383 | 165.378 | 64.0000 | 376.242 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −12.8951 | 16.0000 | 0 | −51.5803 | −48.5387 | 64.0000 | −76.7173 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −0.716545 | 16.0000 | 0 | −2.86618 | −187.375 | 64.0000 | −242.487 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | 9.02921 | 16.0000 | 0 | 36.1168 | 228.688 | 64.0000 | −161.473 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | 27.4670 | 16.0000 | 0 | 109.868 | −1.15327 | 64.0000 | 511.435 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 650.6.a.q | yes | 5 |
| 5.b | even | 2 | 1 | 650.6.a.n | ✓ | 5 | |
| 5.c | odd | 4 | 2 | 650.6.b.n | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.6.a.n | ✓ | 5 | 5.b | even | 2 | 1 | |
| 650.6.a.q | yes | 5 | 1.a | even | 1 | 1 | trivial |
| 650.6.b.n | 10 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + 2T_{3}^{4} - 809T_{3}^{3} - 2922T_{3}^{2} + 77904T_{3} + 57024 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(650))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{5} \)
|
| $3$ |
\( T^{5} + 2 T^{4} + \cdots + 57024 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} - 157 T^{4} + \cdots + 396692544 \)
|
| $11$ |
\( T^{5} + \cdots - 2112867422028 \)
|
| $13$ |
\( (T + 169)^{5} \)
|
| $17$ |
\( T^{5} + \cdots - 84028069477476 \)
|
| $19$ |
\( T^{5} + \cdots + 103203582703616 \)
|
| $23$ |
\( T^{5} + \cdots + 39\!\cdots\!00 \)
|
| $29$ |
\( T^{5} + \cdots + 13\!\cdots\!43 \)
|
| $31$ |
\( T^{5} + \cdots + 80\!\cdots\!72 \)
|
| $37$ |
\( T^{5} + \cdots - 12\!\cdots\!88 \)
|
| $41$ |
\( T^{5} + \cdots - 13\!\cdots\!00 \)
|
| $43$ |
\( T^{5} + \cdots + 61\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots + 49\!\cdots\!88 \)
|
| $53$ |
\( T^{5} + \cdots + 18\!\cdots\!75 \)
|
| $59$ |
\( T^{5} + \cdots - 95\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots - 68\!\cdots\!83 \)
|
| $67$ |
\( T^{5} + \cdots - 11\!\cdots\!72 \)
|
| $71$ |
\( T^{5} + \cdots + 34\!\cdots\!00 \)
|
| $73$ |
\( T^{5} + \cdots + 28\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots + 52\!\cdots\!32 \)
|
| $83$ |
\( T^{5} + \cdots + 20\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots - 15\!\cdots\!76 \)
|
| $97$ |
\( T^{5} + \cdots - 21\!\cdots\!60 \)
|
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