Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1984,4,Mod(1,1984)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1984, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1984.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1984 = 2^{6} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1984.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: | \(117.059789451\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4000044.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 21x^{2} + 16x + 62 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 124) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 21x^{2} + 16x + 62 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 16\nu + 20 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 16\nu - 43 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 21 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 8\beta _1 + 1 \)
|
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 | −8.27238 | 0 | 14.2221 | 0 | 10.5389 | 0 | 41.4323 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.97232 | 0 | −2.58408 | 0 | −13.0539 | 0 | −18.1653 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.74860 | 0 | −20.7669 | 0 | −3.45568 | 0 | −4.45080 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 8.49610 | 0 | 3.12892 | 0 | −10.0292 | 0 | 45.1838 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(31\) | \(1\) |
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 | 1984.4.a.o | 4 | |
| 4.b | odd | 2 | 1 | 1984.4.a.m | 4 | ||
| 8.b | even | 2 | 1 | 496.4.a.f | 4 | ||
| 8.d | odd | 2 | 1 | 124.4.a.b | ✓ | 4 | |
| 24.f | even | 2 | 1 | 1116.4.a.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 124.4.a.b | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 496.4.a.f | 4 | 8.b | even | 2 | 1 | ||
| 1116.4.a.f | 4 | 24.f | even | 2 | 1 | ||
| 1984.4.a.m | 4 | 4.b | odd | 2 | 1 | ||
| 1984.4.a.o | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2T_{3}^{3} - 84T_{3}^{2} + 128T_{3} + 992 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1984))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 992 \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 2388 \)
|
| $7$ |
\( T^{4} + 16 T^{3} + \cdots - 4768 \)
|
| $11$ |
\( T^{4} - 96 T^{3} + \cdots + 16416 \)
|
| $13$ |
\( T^{4} + 42 T^{3} + \cdots + 441648 \)
|
| $17$ |
\( T^{4} + 2 T^{3} + \cdots + 12601872 \)
|
| $19$ |
\( T^{4} - 180 T^{3} + \cdots - 110971856 \)
|
| $23$ |
\( T^{4} + 142 T^{3} + \cdots - 7610112 \)
|
| $29$ |
\( T^{4} + 262 T^{3} + \cdots - 189481680 \)
|
| $31$ |
\( (T + 31)^{4} \)
|
| $37$ |
\( T^{4} - 284 T^{3} + \cdots - 250857520 \)
|
| $41$ |
\( T^{4} + 526 T^{3} + \cdots + 949605156 \)
|
| $43$ |
\( T^{4} + 326 T^{3} + \cdots + 551576736 \)
|
| $47$ |
\( T^{4} + \cdots + 1256256000 \)
|
| $53$ |
\( T^{4} + \cdots - 7248886080 \)
|
| $59$ |
\( T^{4} + 164 T^{3} + \cdots + 42747696 \)
|
| $61$ |
\( T^{4} + \cdots + 37784393520 \)
|
| $67$ |
\( T^{4} + \cdots + 2730691584 \)
|
| $71$ |
\( T^{4} + \cdots + 9642905280 \)
|
| $73$ |
\( T^{4} + \cdots + 231605048496 \)
|
| $79$ |
\( T^{4} + \cdots + 4306806912 \)
|
| $83$ |
\( T^{4} - 1408 T^{3} + \cdots - 747812064 \)
|
| $89$ |
\( T^{4} + \cdots + 25296843024 \)
|
| $97$ |
\( T^{4} + \cdots + 180198106052 \)
|
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