Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1083,4,Mod(1,1083)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1083, 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("1083.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1083 = 3 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1083.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: | \(63.8990685362\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5682368.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 25x^{2} + 24x + 54 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 25x^{2} + 24x + 54 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 13 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 22\nu + 15 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 2\beta_{2} + 24\beta _1 + 11 \)
|
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 |
|
−5.44674 | 3.00000 | 21.6670 | 4.85799 | −16.3402 | 1.11999 | −74.4404 | 9.00000 | −26.4602 | ||||||||||||||||||||||||||||||
| 1.2 | −2.02925 | 3.00000 | −3.88214 | 13.8410 | −6.08775 | 12.9991 | 24.1118 | 9.00000 | −28.0869 | |||||||||||||||||||||||||||||||
| 1.3 | 1.12214 | 3.00000 | −6.74079 | −7.91858 | 3.36643 | 3.25222 | −16.5413 | 9.00000 | −8.88579 | |||||||||||||||||||||||||||||||
| 1.4 | 4.35385 | 3.00000 | 10.9560 | 7.21957 | 13.0615 | −27.3714 | 12.8698 | 9.00000 | 31.4329 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 1083.4.a.e | ✓ | 4 |
| 19.b | odd | 2 | 1 | 1083.4.a.f | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1083.4.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1083.4.a.f | yes | 4 | 19.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 25T_{2}^{2} - 24T_{2} + 54 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1083))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 54 \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 18 T^{3} + \cdots - 3844 \)
|
| $7$ |
\( T^{4} + 10 T^{3} + \cdots - 1296 \)
|
| $11$ |
\( T^{4} + 14 T^{3} + \cdots + 1932772 \)
|
| $13$ |
\( T^{4} + 56 T^{3} + \cdots - 680832 \)
|
| $17$ |
\( T^{4} + 186 T^{3} + \cdots - 10461628 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} + 84 T^{3} + \cdots - 39777856 \)
|
| $29$ |
\( T^{4} - 236 T^{3} + \cdots - 278100864 \)
|
| $31$ |
\( T^{4} + 112 T^{3} + \cdots + 71134848 \)
|
| $37$ |
\( T^{4} + \cdots - 1590786432 \)
|
| $41$ |
\( T^{4} - 68 T^{3} + \cdots + 320141376 \)
|
| $43$ |
\( T^{4} + \cdots - 4236980976 \)
|
| $47$ |
\( T^{4} + \cdots + 4230704988 \)
|
| $53$ |
\( T^{4} + \cdots - 22410191808 \)
|
| $59$ |
\( T^{4} + \cdots + 10491192576 \)
|
| $61$ |
\( T^{4} + \cdots + 50644624356 \)
|
| $67$ |
\( T^{4} + \cdots + 34923875328 \)
|
| $71$ |
\( T^{4} + \cdots + 17183522304 \)
|
| $73$ |
\( T^{4} + \cdots + 13882720932 \)
|
| $79$ |
\( T^{4} + \cdots + 55240890624 \)
|
| $83$ |
\( T^{4} + \cdots + 112165657344 \)
|
| $89$ |
\( T^{4} + \cdots + 25504082496 \)
|
| $97$ |
\( T^{4} + \cdots - 239463482112 \)
|
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