Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [522,4,Mod(1,522)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(522, 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("522.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 522 = 2 \cdot 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 522.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: | \(30.7989970230\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.19816.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 42x - 54 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 58) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 42x - 54 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 4\nu - 27 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 4\beta _1 + 27 \)
|
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 |
|
−2.00000 | 0 | 4.00000 | −20.9205 | 0 | 8.55839 | −8.00000 | 0 | 41.8409 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | −2.36031 | 0 | −18.5045 | −8.00000 | 0 | 4.72062 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 3.28077 | 0 | 33.9461 | −8.00000 | 0 | −6.56153 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(29\) | \(1\) |
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 | 522.4.a.k | 3 | |
| 3.b | odd | 2 | 1 | 58.4.a.d | ✓ | 3 | |
| 12.b | even | 2 | 1 | 464.4.a.i | 3 | ||
| 15.d | odd | 2 | 1 | 1450.4.a.h | 3 | ||
| 24.f | even | 2 | 1 | 1856.4.a.s | 3 | ||
| 24.h | odd | 2 | 1 | 1856.4.a.r | 3 | ||
| 87.d | odd | 2 | 1 | 1682.4.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.4.a.d | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 464.4.a.i | 3 | 12.b | even | 2 | 1 | ||
| 522.4.a.k | 3 | 1.a | even | 1 | 1 | trivial | |
| 1450.4.a.h | 3 | 15.d | odd | 2 | 1 | ||
| 1682.4.a.d | 3 | 87.d | odd | 2 | 1 | ||
| 1856.4.a.r | 3 | 24.h | odd | 2 | 1 | ||
| 1856.4.a.s | 3 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 20T_{5}^{2} - 27T_{5} - 162 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(522))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 20 T^{2} + \cdots - 162 \)
|
| $7$ |
\( T^{3} - 24 T^{2} + \cdots + 5376 \)
|
| $11$ |
\( T^{3} + 10 T^{2} + \cdots - 2424 \)
|
| $13$ |
\( T^{3} + 4 T^{2} + \cdots + 131706 \)
|
| $17$ |
\( T^{3} - 66 T^{2} + \cdots + 679368 \)
|
| $19$ |
\( T^{3} + 164 T^{2} + \cdots - 664448 \)
|
| $23$ |
\( T^{3} - 204 T^{2} + \cdots + 677376 \)
|
| $29$ |
\( (T + 29)^{3} \)
|
| $31$ |
\( T^{3} + 86 T^{2} + \cdots - 4766172 \)
|
| $37$ |
\( T^{3} + 42 T^{2} + \cdots - 7684896 \)
|
| $41$ |
\( T^{3} + 562 T^{2} + \cdots + 5982048 \)
|
| $43$ |
\( T^{3} - 18 T^{2} + \cdots + 196488 \)
|
| $47$ |
\( T^{3} + 654 T^{2} + \cdots + 3425124 \)
|
| $53$ |
\( T^{3} + 712 T^{2} + \cdots - 252120546 \)
|
| $59$ |
\( T^{3} + 184 T^{2} + \cdots - 57362928 \)
|
| $61$ |
\( T^{3} - 322 T^{2} + \cdots - 5254424 \)
|
| $67$ |
\( T^{3} + 228 T^{2} + \cdots + 47608192 \)
|
| $71$ |
\( T^{3} - 52 T^{2} + \cdots + 672 \)
|
| $73$ |
\( T^{3} + 494 T^{2} + \cdots - 9410208 \)
|
| $79$ |
\( T^{3} + 2110 T^{2} + \cdots + 285187172 \)
|
| $83$ |
\( T^{3} - 288 T^{2} + \cdots + 437606064 \)
|
| $89$ |
\( T^{3} + 914 T^{2} + \cdots - 598011552 \)
|
| $97$ |
\( T^{3} - 218 T^{2} + \cdots - 17006112 \)
|
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