Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1449,4,Mod(1,1449)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1449, 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("1449.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1449 = 3^{2} \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1449.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: | \(85.4937675983\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.8184789.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 76x^{2} - 8x + 1048 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 483) |
| 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
\( x^{4} - x^{3} - 76x^{2} - 8x + 1048 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 42\nu + 48 ) / 28 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 3\nu - 38 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 17\nu^{2} + 56\nu - 608 ) / 28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + \beta_{2} + 3\beta _1 + 79 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 51\beta_{3} - 39\beta_{2} + 107\beta _1 + 183 ) / 2 \)
|
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 |
|
−3.70156 | 0 | 5.70156 | 5.05515 | 0 | −7.00000 | 8.50781 | 0 | −18.7119 | ||||||||||||||||||||||||||||||
| 1.2 | −3.70156 | 0 | 5.70156 | 9.64642 | 0 | −7.00000 | 8.50781 | 0 | −35.7068 | |||||||||||||||||||||||||||||||
| 1.3 | 2.70156 | 0 | −0.701562 | −11.0488 | 0 | −7.00000 | −23.5078 | 0 | −29.8490 | |||||||||||||||||||||||||||||||
| 1.4 | 2.70156 | 0 | −0.701562 | 19.3472 | 0 | −7.00000 | −23.5078 | 0 | 52.2678 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 1449.4.a.d | 4 | |
| 3.b | odd | 2 | 1 | 483.4.a.b | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 483.4.a.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 1449.4.a.d | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} - 10 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1449))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T - 10)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 23 T^{3} + \cdots - 10424 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 48 T^{3} + \cdots - 13784 \)
|
| $13$ |
\( T^{4} + 107 T^{3} + \cdots - 12225800 \)
|
| $17$ |
\( T^{4} - 6 T^{3} + \cdots + 2287800 \)
|
| $19$ |
\( T^{4} - 76 T^{3} + \cdots + 58338172 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} + 204 T^{3} + \cdots - 136834868 \)
|
| $31$ |
\( T^{4} + 276 T^{3} + \cdots + 13064256 \)
|
| $37$ |
\( T^{4} + \cdots - 3344244056 \)
|
| $41$ |
\( T^{4} - 450 T^{3} + \cdots + 94298692 \)
|
| $43$ |
\( T^{4} + \cdots + 1500471424 \)
|
| $47$ |
\( T^{4} - 688 T^{3} + \cdots + 369838528 \)
|
| $53$ |
\( T^{4} - 633 T^{3} + \cdots - 180702128 \)
|
| $59$ |
\( T^{4} + 585 T^{3} + \cdots - 69476672 \)
|
| $61$ |
\( T^{4} + \cdots - 29331062568 \)
|
| $67$ |
\( T^{4} + \cdots - 14949193664 \)
|
| $71$ |
\( T^{4} + \cdots + 250630243168 \)
|
| $73$ |
\( T^{4} + \cdots - 333083542356 \)
|
| $79$ |
\( T^{4} + \cdots + 15443424448 \)
|
| $83$ |
\( T^{4} + \cdots - 236900237516 \)
|
| $89$ |
\( T^{4} + \cdots - 642666118448 \)
|
| $97$ |
\( T^{4} + \cdots - 20129702688 \)
|
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