Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1058,4,Mod(1,1058)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1058.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1058 = 2 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1058.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: | \(62.4240207861\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.6584.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 17x - 22 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{3} - 17x - 22 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - \nu - 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 3\nu + 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 22 ) / 2 \)
|
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 | −8.29838 | 4.00000 | 2.95148 | 16.5968 | 29.9194 | −8.00000 | 41.8630 | −5.90295 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 1.23840 | 4.00000 | 9.69062 | −2.47681 | −2.06052 | −8.00000 | −25.4664 | −19.3812 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 5.05997 | 4.00000 | −21.6421 | −10.1199 | 2.14113 | −8.00000 | −1.39668 | 43.2842 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(23\) | \( -1 \) |
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 | 1058.4.a.k | ✓ | 3 |
| 23.b | odd | 2 | 1 | 1058.4.a.l | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1058.4.a.k | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1058.4.a.l | yes | 3 | 23.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):
|
\( T_{3}^{3} + 2T_{3}^{2} - 46T_{3} + 52 \)
|
|
\( T_{5}^{3} + 9T_{5}^{2} - 245T_{5} + 619 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} + 2 T^{2} + \cdots + 52 \)
|
| $5$ |
\( T^{3} + 9 T^{2} + \cdots + 619 \)
|
| $7$ |
\( T^{3} - 30 T^{2} + \cdots + 132 \)
|
| $11$ |
\( T^{3} - 66 T^{2} + \cdots + 138252 \)
|
| $13$ |
\( T^{3} + 23 T^{2} + \cdots - 23167 \)
|
| $17$ |
\( T^{3} + 30 T^{2} + \cdots - 59152 \)
|
| $19$ |
\( T^{3} - 118 T^{2} + \cdots + 119908 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} + 35 T^{2} + \cdots - 1758487 \)
|
| $31$ |
\( T^{3} + 124 T^{2} + \cdots + 2073184 \)
|
| $37$ |
\( T^{3} - 22 T^{2} + \cdots - 1320448 \)
|
| $41$ |
\( T^{3} - 597 T^{2} + \cdots - 5532607 \)
|
| $43$ |
\( T^{3} + 180 T^{2} + \cdots - 8948096 \)
|
| $47$ |
\( T^{3} + 554 T^{2} + \cdots - 42508556 \)
|
| $53$ |
\( T^{3} - 595 T^{2} + \cdots + 15335051 \)
|
| $59$ |
\( T^{3} + 298 T^{2} + \cdots - 141250652 \)
|
| $61$ |
\( T^{3} + 2189 T^{2} + \cdots + 359486903 \)
|
| $67$ |
\( T^{3} - 64 T^{2} + \cdots + 52274304 \)
|
| $71$ |
\( T^{3} + 42 T^{2} + \cdots + 8310756 \)
|
| $73$ |
\( T^{3} + 735 T^{2} + \cdots + 33811125 \)
|
| $79$ |
\( T^{3} + 2504 T^{2} + \cdots + 92731136 \)
|
| $83$ |
\( T^{3} + 240 T^{2} + \cdots - 217015856 \)
|
| $89$ |
\( T^{3} + 1041 T^{2} + \cdots - 180081297 \)
|
| $97$ |
\( T^{3} + 1097 T^{2} + \cdots - 29907553 \)
|
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