Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(21.8121994946\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 129x^{2} - 112x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 129x^{2} - 112x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{2} - 8\nu - 257 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} - 12\nu^{2} - 1026\nu - 419 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 4\beta _1 + 257 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{3} + 9\beta_{2} + 525\beta _1 + 1190 ) / 8 \)
|
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 | −24.5283 | 0 | 49.0560 | 0 | −133.702 | 0 | 358.639 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0.0364806 | 0 | 63.4401 | 0 | −68.3721 | 0 | −242.999 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.72180 | 0 | −32.7534 | 0 | 115.724 | 0 | −240.035 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 20.7701 | 0 | 0.257268 | 0 | −175.650 | 0 | 188.395 | 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 | 136.6.a.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 272.6.a.n | 4 | ||
| 8.b | even | 2 | 1 | 1088.6.a.x | 4 | ||
| 8.d | odd | 2 | 1 | 1088.6.a.w | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.6.a.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 272.6.a.n | 4 | 4.b | odd | 2 | 1 | ||
| 1088.6.a.w | 4 | 8.d | odd | 2 | 1 | ||
| 1088.6.a.x | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{3} - 516T_{3}^{2} + 896T_{3} - 32 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(136))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots - 32 \)
|
| $5$ |
\( T^{4} - 80 T^{3} + \cdots - 26224 \)
|
| $7$ |
\( T^{4} + 262 T^{3} + \cdots - 185818912 \)
|
| $11$ |
\( T^{4} + \cdots + 42499525344 \)
|
| $13$ |
\( T^{4} + \cdots + 68981105328 \)
|
| $17$ |
\( (T - 289)^{4} \)
|
| $19$ |
\( T^{4} + \cdots - 10666408431360 \)
|
| $23$ |
\( T^{4} + \cdots + 45370344432864 \)
|
| $29$ |
\( T^{4} + \cdots - 71489492608752 \)
|
| $31$ |
\( T^{4} + \cdots - 18\!\cdots\!20 \)
|
| $37$ |
\( T^{4} + \cdots - 136195842893232 \)
|
| $41$ |
\( T^{4} + \cdots + 135415199478672 \)
|
| $43$ |
\( T^{4} + \cdots - 16\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots - 25\!\cdots\!64 \)
|
| $53$ |
\( T^{4} + \cdots + 20\!\cdots\!28 \)
|
| $59$ |
\( T^{4} + \cdots + 76\!\cdots\!32 \)
|
| $61$ |
\( T^{4} + \cdots + 91\!\cdots\!80 \)
|
| $67$ |
\( T^{4} + \cdots + 41\!\cdots\!48 \)
|
| $71$ |
\( T^{4} + \cdots + 17\!\cdots\!72 \)
|
| $73$ |
\( T^{4} + \cdots - 61\!\cdots\!32 \)
|
| $79$ |
\( T^{4} + \cdots + 10\!\cdots\!96 \)
|
| $83$ |
\( T^{4} + \cdots - 21\!\cdots\!40 \)
|
| $89$ |
\( T^{4} + \cdots - 25\!\cdots\!32 \)
|
| $97$ |
\( T^{4} + \cdots + 53\!\cdots\!28 \)
|
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