Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1950,4,Mod(1,1950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1950, 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("1950.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1950.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: | \(115.053724511\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 1044x^{2} + 11992x + 24014 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{4} - x^{3} - 1044x^{2} + 11992x + 24014 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} + 91\nu^{2} - 3509\nu - 5639 ) / 181 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 18\nu^{2} + 1281\nu - 18022 ) / 181 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{3} + \beta_{2} - 16\beta _1 + 529 \)
|
| \(\nu^{3}\) | \(=\) |
\( -91\beta_{3} + 18\beta_{2} + 993\beta _1 - 8500 \)
|
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 | −3.00000 | 4.00000 | 0 | 6.00000 | −25.8153 | −8.00000 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.00000 | 4.00000 | 0 | 6.00000 | −23.2511 | −8.00000 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.00000 | 4.00000 | 0 | 6.00000 | −3.25990 | −8.00000 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | −3.00000 | 4.00000 | 0 | 6.00000 | 31.3262 | −8.00000 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1950.4.a.bj | ✓ | 4 |
| 5.b | even | 2 | 1 | 1950.4.a.bq | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1950.4.a.bj | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1950.4.a.bq | yes | 4 | 5.b | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1950))\):
|
\( T_{7}^{4} + 21T_{7}^{3} - 879T_{7}^{2} - 21857T_{7} - 61296 \)
|
|
\( T_{11}^{4} - 33T_{11}^{3} - 1425T_{11}^{2} + 171T_{11} + 24516 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + 21 T^{3} + \cdots - 61296 \)
|
| $11$ |
\( T^{4} - 33 T^{3} + \cdots + 24516 \)
|
| $13$ |
\( (T - 13)^{4} \)
|
| $17$ |
\( T^{4} + 37 T^{3} + \cdots + 28731816 \)
|
| $19$ |
\( T^{4} - 97 T^{3} + \cdots + 107311724 \)
|
| $23$ |
\( T^{4} + 40 T^{3} + \cdots - 28170000 \)
|
| $29$ |
\( T^{4} - 292 T^{3} + \cdots - 49770423 \)
|
| $31$ |
\( T^{4} - 257 T^{3} + \cdots + 59013268 \)
|
| $37$ |
\( T^{4} - 3 T^{3} + \cdots - 551216052 \)
|
| $41$ |
\( T^{4} + \cdots + 3429988740 \)
|
| $43$ |
\( T^{4} + \cdots - 8169330024 \)
|
| $47$ |
\( T^{4} + 676 T^{3} + \cdots - 326221479 \)
|
| $53$ |
\( T^{4} + \cdots + 1071776205 \)
|
| $59$ |
\( T^{4} + 313 T^{3} + \cdots - 501088950 \)
|
| $61$ |
\( T^{4} + \cdots + 44000712068 \)
|
| $67$ |
\( T^{4} + \cdots - 6423757003 \)
|
| $71$ |
\( T^{4} - 303 T^{3} + \cdots - 272876364 \)
|
| $73$ |
\( T^{4} + \cdots + 158848538096 \)
|
| $79$ |
\( T^{4} + \cdots + 47833918652 \)
|
| $83$ |
\( T^{4} + \cdots - 392695327098 \)
|
| $89$ |
\( T^{4} + \cdots + 1495888992 \)
|
| $97$ |
\( T^{4} + \cdots + 115258868000 \)
|
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