Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [43,4,Mod(1,43)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("43.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 43.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: | \(2.53708213025\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.45868.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 10x^{2} + 11x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 10x^{2} + 11x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + \nu^{2} - 8\nu - 2 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + \nu^{2} + 19\nu - 11 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} - \beta _1 + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + 9\beta _1 - 3 \)
|
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 |
|
−5.19893 | 0.759721 | 19.0289 | −1.54763 | −3.94973 | −13.3169 | −57.3382 | −26.4228 | 8.04602 | ||||||||||||||||||||||||||||||
| 1.2 | −1.25341 | 1.08716 | −6.42897 | −14.9226 | −1.36266 | 2.62749 | 18.0854 | −25.8181 | 18.7041 | |||||||||||||||||||||||||||||||
| 1.3 | 0.132290 | −3.71054 | −7.98250 | 2.43196 | −0.490867 | −30.9271 | −2.11433 | −13.2319 | 0.321724 | |||||||||||||||||||||||||||||||
| 1.4 | 2.32005 | −9.13635 | −2.61739 | −12.9617 | −21.1967 | 21.6165 | −24.6328 | 56.4728 | −30.0718 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(43\) | \(-1\) |
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 | 43.4.a.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 387.4.a.e | 4 | ||
| 4.b | odd | 2 | 1 | 688.4.a.f | 4 | ||
| 5.b | even | 2 | 1 | 1075.4.a.a | 4 | ||
| 7.b | odd | 2 | 1 | 2107.4.a.b | 4 | ||
| 43.b | odd | 2 | 1 | 1849.4.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 43.4.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 387.4.a.e | 4 | 3.b | odd | 2 | 1 | ||
| 688.4.a.f | 4 | 4.b | odd | 2 | 1 | ||
| 1075.4.a.a | 4 | 5.b | even | 2 | 1 | ||
| 1849.4.a.b | 4 | 43.b | odd | 2 | 1 | ||
| 2107.4.a.b | 4 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 9T_{2}^{2} - 14T_{2} + 2 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(43))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
|
| $3$ |
\( T^{4} + 11 T^{3} + \cdots + 28 \)
|
| $5$ |
\( T^{4} + 27 T^{3} + \cdots - 728 \)
|
| $7$ |
\( T^{4} + 20 T^{3} + \cdots + 23392 \)
|
| $11$ |
\( T^{4} + 62 T^{3} + \cdots - 3329968 \)
|
| $13$ |
\( T^{4} + 2 T^{3} + \cdots - 7252 \)
|
| $17$ |
\( T^{4} + 207 T^{3} + \cdots - 162866753 \)
|
| $19$ |
\( T^{4} - 99 T^{3} + \cdots + 12456052 \)
|
| $23$ |
\( T^{4} + 103 T^{3} + \cdots + 62171919 \)
|
| $29$ |
\( T^{4} + \cdots + 2739631572 \)
|
| $31$ |
\( T^{4} - 131 T^{3} + \cdots - 113939343 \)
|
| $37$ |
\( T^{4} + 449 T^{3} + \cdots + 43975584 \)
|
| $41$ |
\( T^{4} + 491 T^{3} + \cdots - 226001461 \)
|
| $43$ |
\( (T - 43)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 10139289552 \)
|
| $53$ |
\( T^{4} + \cdots - 7756027948 \)
|
| $59$ |
\( T^{4} - 816 T^{3} + \cdots + 125277376 \)
|
| $61$ |
\( T^{4} - 372 T^{3} + \cdots - 912187696 \)
|
| $67$ |
\( T^{4} + \cdots - 1865644524 \)
|
| $71$ |
\( T^{4} - 468 T^{3} + \cdots + 983961904 \)
|
| $73$ |
\( T^{4} + \cdots + 53103596112 \)
|
| $79$ |
\( T^{4} + \cdots + 41301634784 \)
|
| $83$ |
\( T^{4} + \cdots - 243170481652 \)
|
| $89$ |
\( T^{4} + \cdots + 93359062944 \)
|
| $97$ |
\( T^{4} + \cdots - 82907544713 \)
|
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