Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [87,4,Mod(1,87)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("87.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 87 = 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 87.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: | \(5.13316617050\) |
| 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: |
\( x^{5} - 2x^{4} - 32x^{3} + 21x^{2} + 222x + 32 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{5} - 2x^{4} - 32x^{3} + 21x^{2} + 222x + 32 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 22\nu^{2} - 12\nu + 23 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 33\nu^{2} - 10\nu - 166 ) / 11 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -3\nu^{4} + 14\nu^{3} + 66\nu^{2} - 195\nu - 300 ) / 22 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 3\beta_{2} + 21\beta _1 + 21 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 22\beta_{3} + 36\beta_{2} + 77\beta _1 + 284 \)
|
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 |
|
−4.73479 | 3.00000 | 14.4183 | 21.2830 | −14.2044 | −25.6129 | −30.3891 | 9.00000 | −100.771 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.29334 | 3.00000 | −2.74060 | −1.31275 | −6.88002 | 6.75369 | 24.6318 | 9.00000 | 3.01058 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.14663 | 3.00000 | −6.68524 | 8.47253 | 3.43988 | 26.2528 | −16.8385 | 9.00000 | 9.71484 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.90829 | 3.00000 | 7.27471 | 11.0111 | 11.7249 | −6.56501 | −2.83463 | 9.00000 | 43.0344 | ||||||||||||||||||||||||||||||||||
| 1.5 | 4.97322 | 3.00000 | 16.7329 | −10.4539 | 14.9196 | 3.17147 | 43.4304 | 9.00000 | −51.9893 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(29\) | \(-1\) |
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 | 87.4.a.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 261.4.a.d | 5 | ||
| 4.b | odd | 2 | 1 | 1392.4.a.s | 5 | ||
| 5.b | even | 2 | 1 | 2175.4.a.h | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 87.4.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 261.4.a.d | 5 | 3.b | odd | 2 | 1 | ||
| 1392.4.a.s | 5 | 4.b | odd | 2 | 1 | ||
| 2175.4.a.h | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 3T_{2}^{4} - 30T_{2}^{3} + 77T_{2}^{2} + 165T_{2} - 242 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(87))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 3 T^{4} + \cdots - 242 \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( T^{5} - 29 T^{4} + \cdots - 27248 \)
|
| $7$ |
\( T^{5} - 4 T^{4} + \cdots - 94552 \)
|
| $11$ |
\( T^{5} - 69 T^{4} + \cdots + 11227720 \)
|
| $13$ |
\( T^{5} - 29 T^{4} + \cdots - 108021296 \)
|
| $17$ |
\( T^{5} + \cdots - 4937206022 \)
|
| $19$ |
\( T^{5} + 47 T^{4} + \cdots - 138008160 \)
|
| $23$ |
\( T^{5} + \cdots + 29685745664 \)
|
| $29$ |
\( (T - 29)^{5} \)
|
| $31$ |
\( T^{5} + 318 T^{4} + \cdots + 78718976 \)
|
| $37$ |
\( T^{5} + \cdots + 289102067936 \)
|
| $41$ |
\( T^{5} + \cdots - 846879647600 \)
|
| $43$ |
\( T^{5} + 333 T^{4} + \cdots + 189276160 \)
|
| $47$ |
\( T^{5} + \cdots + 421428897792 \)
|
| $53$ |
\( T^{5} + \cdots + 5034370202272 \)
|
| $59$ |
\( T^{5} + \cdots - 490681070624 \)
|
| $61$ |
\( T^{5} + \cdots + 25227106613632 \)
|
| $67$ |
\( T^{5} + \cdots + 2252072401984 \)
|
| $71$ |
\( T^{5} + \cdots - 47926056484864 \)
|
| $73$ |
\( T^{5} + \cdots - 66833927347904 \)
|
| $79$ |
\( T^{5} + \cdots + 4966584868352 \)
|
| $83$ |
\( T^{5} + \cdots - 140945035365376 \)
|
| $89$ |
\( T^{5} + \cdots - 4931164609420 \)
|
| $97$ |
\( T^{5} + \cdots - 881915066000000 \)
|
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