Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,5,Mod(115,136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(136, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.115");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 136.j (of order \(4\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(14.0583149794\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( 8\zeta_{8} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\zeta_{8}^{3} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_1 ) / 8 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( \beta_{3} ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
| \(n\) | \(69\) | \(103\) | \(105\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 115.1 |
|
− | 4.00000i | 1.34315 | + | 1.34315i | −16.0000 | 0 | 5.37258 | − | 5.37258i | 0 | 64.0000i | − | 77.3919i | 0 | ||||||||||||||||||||||||
| 115.2 | − | 4.00000i | 12.6569 | + | 12.6569i | −16.0000 | 0 | 50.6274 | − | 50.6274i | 0 | 64.0000i | 239.392i | 0 | ||||||||||||||||||||||||||
| 123.1 | 4.00000i | 1.34315 | − | 1.34315i | −16.0000 | 0 | 5.37258 | + | 5.37258i | 0 | − | 64.0000i | 77.3919i | 0 | ||||||||||||||||||||||||||
| 123.2 | 4.00000i | 12.6569 | − | 12.6569i | −16.0000 | 0 | 50.6274 | + | 50.6274i | 0 | − | 64.0000i | − | 239.392i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 17.c | even | 4 | 1 | inner |
| 136.j | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 136.5.j.a | ✓ | 4 |
| 8.d | odd | 2 | 1 | CM | 136.5.j.a | ✓ | 4 |
| 17.c | even | 4 | 1 | inner | 136.5.j.a | ✓ | 4 |
| 136.j | odd | 4 | 1 | inner | 136.5.j.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.5.j.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 136.5.j.a | ✓ | 4 | 8.d | odd | 2 | 1 | CM |
| 136.5.j.a | ✓ | 4 | 17.c | even | 4 | 1 | inner |
| 136.5.j.a | ✓ | 4 | 136.j | odd | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 28T_{3}^{3} + 392T_{3}^{2} - 952T_{3} + 1156 \)
acting on \(S_{5}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16)^{2} \)
|
| $3$ |
\( T^{4} - 28 T^{3} + \cdots + 1156 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 92 T^{3} + \cdots + 737991556 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 6975757441 \)
|
| $19$ |
\( (T^{2} + 188356)^{2} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 16801850188036 \)
|
| $43$ |
\( (T^{2} + 1411200)^{2} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( (T^{2} + 48412800)^{2} \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T^{2} - 54246528)^{2} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( (T^{2} + 64706688)^{2} \)
|
| $89$ |
\( (T^{2} - 221004288)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 59\!\cdots\!44 \)
|
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