Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [798,4,Mod(1,798)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(798, 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("798.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 798.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: | \(47.0835241846\) |
| 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: |
\( x^{4} - 149x^{2} - 30x + 190 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - 149x^{2} - 30x + 190 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} - \nu^{2} + 287\nu + 109 ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} + 133\nu - 733 ) / 21 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} - 2\beta_{2} + \beta _1 + 150 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} - 20\beta_{2} + 143\beta _1 + 34 ) / 2 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | −8.66638 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | 17.3328 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −6.76090 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | 13.5218 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 3.22917 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −6.45833 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 3.00000 | 4.00000 | 22.1981 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −44.3962 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(19\) | \(-1\) |
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 | 798.4.a.n | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2394.4.a.t | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 798.4.a.n | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 2394.4.a.t | 4 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 10T_{5}^{3} - 262T_{5}^{2} - 384T_{5} + 4200 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(798))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} - 10 T^{3} + \cdots + 4200 \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 64 T^{3} + \cdots + 121600 \)
|
| $13$ |
\( T^{4} - 16 T^{3} + \cdots - 1676640 \)
|
| $17$ |
\( T^{4} + 78 T^{3} + \cdots + 1694728 \)
|
| $19$ |
\( (T - 19)^{4} \)
|
| $23$ |
\( T^{4} - 64 T^{3} + \cdots - 6139136 \)
|
| $29$ |
\( T^{4} - 70 T^{3} + \cdots + 158392440 \)
|
| $31$ |
\( T^{4} + 28 T^{3} + \cdots - 197585088 \)
|
| $37$ |
\( T^{4} - 224 T^{3} + \cdots + 204895888 \)
|
| $41$ |
\( T^{4} + \cdots - 1300301424 \)
|
| $43$ |
\( T^{4} - 348 T^{3} + \cdots + 612419520 \)
|
| $47$ |
\( T^{4} + \cdots - 18589558304 \)
|
| $53$ |
\( T^{4} + \cdots + 18470430200 \)
|
| $59$ |
\( T^{4} + \cdots - 26797448192 \)
|
| $61$ |
\( T^{4} + \cdots + 12073242512 \)
|
| $67$ |
\( T^{4} + \cdots + 11549979840 \)
|
| $71$ |
\( T^{4} - 1154 T^{3} + \cdots - 117560000 \)
|
| $73$ |
\( T^{4} + \cdots - 59607087088 \)
|
| $79$ |
\( T^{4} + \cdots - 325127704448 \)
|
| $83$ |
\( T^{4} + \cdots + 190619676640 \)
|
| $89$ |
\( T^{4} + \cdots + 18924333840 \)
|
| $97$ |
\( T^{4} + \cdots - 685563416080 \)
|
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