Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [930,4,Mod(1,930)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, 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("930.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 930.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: | \(54.8717763053\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.54324.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 36x - 70 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 36x - 70 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3\beta _1 + 24 \)
|
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 | −5.00000 | −6.00000 | −16.5833 | −8.00000 | 9.00000 | 10.0000 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | 1.09816 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | −5.00000 | −6.00000 | 15.4851 | −8.00000 | 9.00000 | 10.0000 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(31\) | \(1\) |
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 | 930.4.a.f | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 930.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 258T_{7} + 282 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(930))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( T^{3} - 258T + 282 \)
|
| $11$ |
\( T^{3} - 18 T^{2} + \cdots - 1240 \)
|
| $13$ |
\( T^{3} + 36 T^{2} + \cdots - 124494 \)
|
| $17$ |
\( T^{3} - 72 T^{2} + \cdots + 71660 \)
|
| $19$ |
\( T^{3} + 168 T^{2} + \cdots - 391424 \)
|
| $23$ |
\( T^{3} + 42 T^{2} + \cdots + 290780 \)
|
| $29$ |
\( T^{3} - 42 T^{2} + \cdots - 3383434 \)
|
| $31$ |
\( (T + 31)^{3} \)
|
| $37$ |
\( T^{3} + 54 T^{2} + \cdots - 3213166 \)
|
| $41$ |
\( T^{3} - 288 T^{2} + \cdots - 672000 \)
|
| $43$ |
\( T^{3} + 426 T^{2} + \cdots - 65357000 \)
|
| $47$ |
\( T^{3} + 234 T^{2} + \cdots - 29110116 \)
|
| $53$ |
\( T^{3} - 516 T^{2} + \cdots + 14062340 \)
|
| $59$ |
\( T^{3} + 336 T^{2} + \cdots - 25255390 \)
|
| $61$ |
\( T^{3} + 1266 T^{2} + \cdots + 59201400 \)
|
| $67$ |
\( T^{3} + 1122 T^{2} + \cdots + 51883298 \)
|
| $71$ |
\( T^{3} + 1146 T^{2} + \cdots + 20572530 \)
|
| $73$ |
\( T^{3} + 1014 T^{2} + \cdots + 4997934 \)
|
| $79$ |
\( T^{3} + 1044 T^{2} + \cdots - 97106996 \)
|
| $83$ |
\( T^{3} - 414 T^{2} + \cdots + 939990468 \)
|
| $89$ |
\( T^{3} - 60 T^{2} + \cdots + 163143870 \)
|
| $97$ |
\( T^{3} + 492 T^{2} + \cdots + 446143040 \)
|
| show more | |
| show less |