Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [552,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("552.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 552 = 2^{3} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(32.5690543232\) |
| 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: |
\( x^{5} - 116x^{3} - 489x^{2} - 48x + 864 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{5} - 116x^{3} - 489x^{2} - 48x + 864 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -47\nu^{4} + 72\nu^{3} + 4900\nu^{2} + 20247\nu + 6708 ) / 2076 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -67\nu^{4} + 456\nu^{3} + 5660\nu^{2} - 9477\nu - 33636 ) / 2076 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -33\nu^{4} + 80\nu^{3} + 3676\nu^{2} + 7561\nu - 15152 ) / 692 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 47\nu^{4} - 72\nu^{3} - 4900\nu^{2} - 16095\nu - 6708 ) / 1038 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 + 94 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 73\beta_{4} + 15\beta_{3} + 8\beta_{2} + 103\beta _1 + 597 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 1697\beta_{4} + 880\beta_{3} - 184\beta_{2} + 1626\beta _1 + 22000 ) / 4 \)
|
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 | −3.00000 | 0 | −16.4663 | 0 | 14.5868 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −12.0612 | 0 | −0.637167 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 6.15949 | 0 | −29.7910 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 8.19322 | 0 | −2.65830 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −3.00000 | 0 | 18.1748 | 0 | 4.49976 | 0 | 9.00000 | 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 | 552.4.a.i | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1656.4.a.p | 5 | ||
| 4.b | odd | 2 | 1 | 1104.4.a.x | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 552.4.a.i | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1104.4.a.x | 5 | 4.b | odd | 2 | 1 | ||
| 1656.4.a.p | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 4T_{5}^{4} - 418T_{5}^{3} + 1504T_{5}^{2} + 35664T_{5} - 182160 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(552))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( T^{5} - 4 T^{4} + \cdots - 182160 \)
|
| $7$ |
\( T^{5} + 14 T^{4} + \cdots + 3312 \)
|
| $11$ |
\( T^{5} + 20 T^{4} + \cdots + 11639808 \)
|
| $13$ |
\( T^{5} - 38 T^{4} + \cdots - 190195328 \)
|
| $17$ |
\( T^{5} + \cdots - 2496028416 \)
|
| $19$ |
\( T^{5} + \cdots + 1332638400 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 123117039840 \)
|
| $31$ |
\( T^{5} + \cdots + 163219952640 \)
|
| $37$ |
\( T^{5} + \cdots - 955239441152 \)
|
| $41$ |
\( T^{5} + \cdots - 2226218422560 \)
|
| $43$ |
\( T^{5} + \cdots + 50378242144 \)
|
| $47$ |
\( T^{5} + \cdots - 24806719928832 \)
|
| $53$ |
\( T^{5} + \cdots - 3862849374480 \)
|
| $59$ |
\( T^{5} + \cdots + 152754933504 \)
|
| $61$ |
\( T^{5} + \cdots + 2497468850624 \)
|
| $67$ |
\( T^{5} + \cdots + 97983371434848 \)
|
| $71$ |
\( T^{5} + \cdots - 155189902573568 \)
|
| $73$ |
\( T^{5} + \cdots + 100620789362400 \)
|
| $79$ |
\( T^{5} + \cdots + 8032929968592 \)
|
| $83$ |
\( T^{5} + \cdots + 33272999123328 \)
|
| $89$ |
\( T^{5} + \cdots - 1839853328640 \)
|
| $97$ |
\( T^{5} + \cdots + 171571831005792 \)
|
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