Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,4,Mod(1,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, 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("136.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 136.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: | \(8.02425976078\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.8396.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{3} - x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 14 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{2} + \beta _1 + 29 ) / 2 \)
|
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 |
|
0 | −8.11952 | 0 | 11.0296 | 0 | 5.74786 | 0 | 38.9265 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.16862 | 0 | 1.28534 | 0 | −30.0364 | 0 | −25.6343 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 2.95089 | 0 | −20.3149 | 0 | 22.2885 | 0 | −18.2922 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
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 | 136.4.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1224.4.a.i | 3 | ||
| 4.b | odd | 2 | 1 | 272.4.a.j | 3 | ||
| 8.b | even | 2 | 1 | 1088.4.a.y | 3 | ||
| 8.d | odd | 2 | 1 | 1088.4.a.u | 3 | ||
| 12.b | even | 2 | 1 | 2448.4.a.bj | 3 | ||
| 17.b | even | 2 | 1 | 2312.4.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.4.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 272.4.a.j | 3 | 4.b | odd | 2 | 1 | ||
| 1088.4.a.u | 3 | 8.d | odd | 2 | 1 | ||
| 1088.4.a.y | 3 | 8.b | even | 2 | 1 | ||
| 1224.4.a.i | 3 | 3.b | odd | 2 | 1 | ||
| 2312.4.a.d | 3 | 17.b | even | 2 | 1 | ||
| 2448.4.a.bj | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 4T_{3}^{2} - 30T_{3} + 28 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(136))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 4 T^{2} + \cdots + 28 \)
|
| $5$ |
\( T^{3} + 8 T^{2} + \cdots + 288 \)
|
| $7$ |
\( T^{3} + 2 T^{2} + \cdots + 3848 \)
|
| $11$ |
\( T^{3} + 84 T^{2} + \cdots + 10972 \)
|
| $13$ |
\( T^{3} + 50 T^{2} + \cdots - 16048 \)
|
| $17$ |
\( (T + 17)^{3} \)
|
| $19$ |
\( T^{3} + 224 T^{2} + \cdots + 322592 \)
|
| $23$ |
\( T^{3} + 234 T^{2} + \cdots + 463504 \)
|
| $29$ |
\( T^{3} + 72 T^{2} + \cdots - 1862624 \)
|
| $31$ |
\( T^{3} + 2 T^{2} + \cdots - 1252344 \)
|
| $37$ |
\( T^{3} - 100 T^{2} + \cdots - 4014352 \)
|
| $41$ |
\( T^{3} - 218 T^{2} + \cdots + 7651496 \)
|
| $43$ |
\( T^{3} - 44 T^{2} + \cdots - 18647936 \)
|
| $47$ |
\( T^{3} + 16 T^{2} + \cdots - 7663872 \)
|
| $53$ |
\( T^{3} - 462 T^{2} + \cdots + 10502936 \)
|
| $59$ |
\( T^{3} + 68 T^{2} + \cdots - 81654208 \)
|
| $61$ |
\( T^{3} - 460 T^{2} + \cdots + 116249648 \)
|
| $67$ |
\( T^{3} + 1008 T^{2} + \cdots - 534256128 \)
|
| $71$ |
\( T^{3} - 518 T^{2} + \cdots + 93696432 \)
|
| $73$ |
\( T^{3} - 838 T^{2} + \cdots + 324704504 \)
|
| $79$ |
\( T^{3} - 1238 T^{2} + \cdots - 7088608 \)
|
| $83$ |
\( T^{3} + 1148 T^{2} + \cdots - 158784832 \)
|
| $89$ |
\( T^{3} + 2506 T^{2} + \cdots + 456757648 \)
|
| $97$ |
\( T^{3} + \cdots + 1961875944 \)
|
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