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: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.123880.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 37x - 54 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{3} - 37x - 54 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 25 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 25 \)
|
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 | −2.14904 | 4.00000 | −10.8107 | −4.29808 | −3.66165 | 8.00000 | −22.3816 | −21.6214 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | 1.43743 | 4.00000 | 20.4332 | 2.87485 | 23.9958 | 8.00000 | −24.9338 | 40.8664 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 9.71161 | 4.00000 | −5.62253 | 19.4232 | −10.3341 | 8.00000 | 67.3154 | −11.2451 | ||||||||||||||||||||||||||||
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.n | yes | 3 |
| 23.b | odd | 2 | 1 | 1058.4.a.m | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1058.4.a.m | ✓ | 3 | 23.b | odd | 2 | 1 | |
| 1058.4.a.n | yes | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1058))\):
|
\( T_{3}^{3} - 9T_{3}^{2} - 10T_{3} + 30 \)
|
|
\( T_{5}^{3} - 4T_{5}^{2} - 275T_{5} - 1242 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
|
| $3$ |
\( T^{3} - 9 T^{2} + \cdots + 30 \)
|
| $5$ |
\( T^{3} - 4 T^{2} + \cdots - 1242 \)
|
| $7$ |
\( T^{3} - 10 T^{2} + \cdots - 908 \)
|
| $11$ |
\( T^{3} + 8 T^{2} + \cdots + 11448 \)
|
| $13$ |
\( T^{3} - 46 T^{2} + \cdots + 33772 \)
|
| $17$ |
\( T^{3} - 95 T^{2} + \cdots + 967464 \)
|
| $19$ |
\( T^{3} + 9 T^{2} + \cdots + 200646 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} + 22 T^{2} + \cdots - 3768588 \)
|
| $31$ |
\( T^{3} - 454 T^{2} + \cdots - 2812440 \)
|
| $37$ |
\( T^{3} - 710 T^{2} + \cdots - 6473664 \)
|
| $41$ |
\( T^{3} + 380 T^{2} + \cdots - 714150 \)
|
| $43$ |
\( T^{3} + 459 T^{2} + \cdots - 33905148 \)
|
| $47$ |
\( T^{3} - 262 T^{2} + \cdots + 32736636 \)
|
| $53$ |
\( T^{3} + 610 T^{2} + \cdots + 4339824 \)
|
| $59$ |
\( T^{3} - 9 T^{2} + \cdots - 3034422 \)
|
| $61$ |
\( T^{3} - 158 T^{2} + \cdots + 5550140 \)
|
| $67$ |
\( T^{3} + 487 T^{2} + \cdots - 7065456 \)
|
| $71$ |
\( T^{3} - 738 T^{2} + \cdots + 34449300 \)
|
| $73$ |
\( T^{3} - 1141 T^{2} + \cdots + 75160161 \)
|
| $79$ |
\( T^{3} + 212 T^{2} + \cdots - 355703616 \)
|
| $83$ |
\( T^{3} - 1049 T^{2} + \cdots + 82133364 \)
|
| $89$ |
\( T^{3} + 2328 T^{2} + \cdots + 246647430 \)
|
| $97$ |
\( T^{3} - 2548 T^{2} + \cdots - 475476966 \)
|
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