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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.93944.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 72x + 44 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 72x + 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 50 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{2} - \beta _1 + 99 ) / 2 \)
|
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 | −5.58370 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | 11.1674 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.00000 | 4.00000 | 0.328719 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | −0.657438 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.00000 | 4.00000 | 15.2550 | 6.00000 | 7.00000 | −8.00000 | 9.00000 | −30.5100 | ||||||||||||||||||||||||||||
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.f | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2394.4.a.n | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 798.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2394.4.a.n | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 10T_{5}^{2} - 82T_{5} + 28 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(798))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 10 T^{2} + \cdots + 28 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} - 1256T + 15808 \)
|
| $13$ |
\( T^{3} + 50 T^{2} + \cdots - 352 \)
|
| $17$ |
\( T^{3} - 18 T^{2} + \cdots + 11564 \)
|
| $19$ |
\( (T + 19)^{3} \)
|
| $23$ |
\( T^{3} + 40 T^{2} + \cdots - 401152 \)
|
| $29$ |
\( T^{3} + 54 T^{2} + \cdots - 3203764 \)
|
| $31$ |
\( T^{3} + 336 T^{2} + \cdots - 9870992 \)
|
| $37$ |
\( T^{3} + 282 T^{2} + \cdots - 178632 \)
|
| $41$ |
\( T^{3} - 150 T^{2} + \cdots + 4299592 \)
|
| $43$ |
\( T^{3} - 184 T^{2} + \cdots + 14097776 \)
|
| $47$ |
\( T^{3} - 708 T^{2} + \cdots - 1033784 \)
|
| $53$ |
\( T^{3} - 378 T^{2} + \cdots + 1396124 \)
|
| $59$ |
\( T^{3} - 260 T^{2} + \cdots + 74998656 \)
|
| $61$ |
\( T^{3} - 90 T^{2} + \cdots + 22763752 \)
|
| $67$ |
\( T^{3} + 280 T^{2} + \cdots - 16792448 \)
|
| $71$ |
\( T^{3} + 248 T^{2} + \cdots - 13951648 \)
|
| $73$ |
\( T^{3} + 978 T^{2} + \cdots - 160034168 \)
|
| $79$ |
\( T^{3} + 664 T^{2} + \cdots + 35746816 \)
|
| $83$ |
\( T^{3} - 516 T^{2} + \cdots + 840387096 \)
|
| $89$ |
\( T^{3} + 90 T^{2} + \cdots - 515242616 \)
|
| $97$ |
\( T^{3} + 1390 T^{2} + \cdots - 116520184 \)
|
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