Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [552,6,Mod(1,552)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(552, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("552.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 552.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: | \(88.5318685368\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{5} - 2x^{4} - 374x^{3} + 1565x^{2} + 19136x - 84640 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 374x^{3} + 1565x^{2} + 19136x - 84640 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 282\nu^{2} + 1289\nu + 1541 ) / 207 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 558\nu^{2} - 2669\nu - 42872 ) / 828 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 374\nu^{2} - 93\nu - 16008 ) / 276 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 25\nu^{4} + 134\nu^{3} - 8430\nu^{2} - 14971\nu + 355396 ) / 2484 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{3} - 2\beta_{2} + \beta _1 + 5 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{3} + 13\beta_{2} + 7\beta _1 + 911 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 27\beta_{4} + 98\beta_{3} - 53\beta_{2} + 4\beta _1 - 953 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 324\beta_{4} + 1418\beta_{3} + 9274\beta_{2} + 5191\beta _1 + 477431 ) / 12 \)
|
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 | 9.00000 | 0 | −53.5417 | 0 | −8.43952 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −46.0012 | 0 | −118.619 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 9.00000 | 0 | 5.87631 | 0 | 170.213 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.00000 | 0 | 48.9489 | 0 | −70.3263 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 9.00000 | 0 | 60.7177 | 0 | −106.828 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 552.6.a.b | ✓ | 5 |
| 4.b | odd | 2 | 1 | 1104.6.a.q | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 552.6.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1104.6.a.q | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 16T_{5}^{4} - 5422T_{5}^{3} + 57952T_{5}^{2} + 7168896T_{5} - 43015536 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(552))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T - 9)^{5} \)
|
| $5$ |
\( T^{5} - 16 T^{4} + \cdots - 43015536 \)
|
| $7$ |
\( T^{5} + \cdots - 1280163600 \)
|
| $11$ |
\( T^{5} + \cdots - 434252710656 \)
|
| $13$ |
\( T^{5} + \cdots - 554621060864 \)
|
| $17$ |
\( T^{5} + \cdots - 37502816241024 \)
|
| $19$ |
\( T^{5} + \cdots + 457300907059680 \)
|
| $23$ |
\( (T - 529)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 795792462867936 \)
|
| $31$ |
\( T^{5} + \cdots + 86\!\cdots\!60 \)
|
| $37$ |
\( T^{5} + \cdots + 22\!\cdots\!36 \)
|
| $41$ |
\( T^{5} + \cdots - 15\!\cdots\!64 \)
|
| $43$ |
\( T^{5} + \cdots - 17\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 36\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $59$ |
\( T^{5} + \cdots - 43\!\cdots\!44 \)
|
| $61$ |
\( T^{5} + \cdots + 12\!\cdots\!20 \)
|
| $67$ |
\( T^{5} + \cdots + 37\!\cdots\!12 \)
|
| $71$ |
\( T^{5} + \cdots + 20\!\cdots\!92 \)
|
| $73$ |
\( T^{5} + \cdots - 16\!\cdots\!20 \)
|
| $79$ |
\( T^{5} + \cdots + 23\!\cdots\!24 \)
|
| $83$ |
\( T^{5} + \cdots + 98\!\cdots\!64 \)
|
| $89$ |
\( T^{5} + \cdots - 14\!\cdots\!20 \)
|
| $97$ |
\( T^{5} + \cdots - 14\!\cdots\!08 \)
|
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