Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [136,2,Mod(13,136)]
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([0, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("136.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 136 = 2^{3} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 136.i (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: | \(1.08596546749\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.342102016.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 3\nu^{4} + 2\nu^{2} + 8\nu + 8 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{3} + 2\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{5} + 4\nu^{3} + 4\nu ) / 8 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} - \nu^{4} + 6\nu^{2} + 8\nu ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{6} - \nu^{4} - 4\nu^{2} - 4 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + 2\beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 4\beta_{4} + 2\beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{7} - 3\beta_{6} + 2\beta_{3} + 3\beta _1 - 4 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -3\beta_{7} - \beta_{6} - 10\beta_{3} + \beta _1 - 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + 6\beta_{5} - 20\beta_{4} - 2\beta_{2} - \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).
| \(n\) | \(69\) | \(103\) | \(105\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
−1.17915 | − | 0.780776i | 0.662153 | − | 0.662153i | 0.780776 | + | 1.84130i | 0 | −1.29777 | + | 0.263783i | 2.56155 | − | 2.56155i | 0.516994 | − | 2.78078i | 2.12311i | 0 | ||||||||||||||||||||||||||||||
| 13.2 | −0.599676 | + | 1.28078i | −2.13578 | + | 2.13578i | −1.28078 | − | 1.53610i | 0 | −1.45468 | − | 4.01623i | −1.56155 | + | 1.56155i | 2.73546 | − | 0.719224i | − | 6.12311i | 0 | ||||||||||||||||||||||||||||||
| 13.3 | 0.599676 | + | 1.28078i | 2.13578 | − | 2.13578i | −1.28078 | + | 1.53610i | 0 | 4.01623 | + | 1.45468i | −1.56155 | + | 1.56155i | −2.73546 | − | 0.719224i | − | 6.12311i | 0 | ||||||||||||||||||||||||||||||
| 13.4 | 1.17915 | − | 0.780776i | −0.662153 | + | 0.662153i | 0.780776 | − | 1.84130i | 0 | −0.263783 | + | 1.29777i | 2.56155 | − | 2.56155i | −0.516994 | − | 2.78078i | 2.12311i | 0 | |||||||||||||||||||||||||||||||
| 21.1 | −1.17915 | + | 0.780776i | 0.662153 | + | 0.662153i | 0.780776 | − | 1.84130i | 0 | −1.29777 | − | 0.263783i | 2.56155 | + | 2.56155i | 0.516994 | + | 2.78078i | − | 2.12311i | 0 | ||||||||||||||||||||||||||||||
| 21.2 | −0.599676 | − | 1.28078i | −2.13578 | − | 2.13578i | −1.28078 | + | 1.53610i | 0 | −1.45468 | + | 4.01623i | −1.56155 | − | 1.56155i | 2.73546 | + | 0.719224i | 6.12311i | 0 | |||||||||||||||||||||||||||||||
| 21.3 | 0.599676 | − | 1.28078i | 2.13578 | + | 2.13578i | −1.28078 | − | 1.53610i | 0 | 4.01623 | − | 1.45468i | −1.56155 | − | 1.56155i | −2.73546 | + | 0.719224i | 6.12311i | 0 | |||||||||||||||||||||||||||||||
| 21.4 | 1.17915 | + | 0.780776i | −0.662153 | − | 0.662153i | 0.780776 | + | 1.84130i | 0 | −0.263783 | − | 1.29777i | 2.56155 | + | 2.56155i | −0.516994 | + | 2.78078i | − | 2.12311i | 0 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
| 17.c | even | 4 | 1 | inner |
| 136.i | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 136.2.i.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 544.2.m.a | 8 | ||
| 8.b | even | 2 | 1 | inner | 136.2.i.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | 544.2.m.a | 8 | ||
| 17.c | even | 4 | 1 | inner | 136.2.i.a | ✓ | 8 |
| 68.f | odd | 4 | 1 | 544.2.m.a | 8 | ||
| 136.i | even | 4 | 1 | inner | 136.2.i.a | ✓ | 8 |
| 136.j | odd | 4 | 1 | 544.2.m.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.2.i.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 136.2.i.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 136.2.i.a | ✓ | 8 | 17.c | even | 4 | 1 | inner |
| 136.2.i.a | ✓ | 8 | 136.i | even | 4 | 1 | inner |
| 544.2.m.a | 8 | 4.b | odd | 2 | 1 | ||
| 544.2.m.a | 8 | 8.d | odd | 2 | 1 | ||
| 544.2.m.a | 8 | 68.f | odd | 4 | 1 | ||
| 544.2.m.a | 8 | 136.j | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 84T_{3}^{4} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{6} + \cdots + 16 \)
|
| $3$ |
\( T^{8} + 84T^{4} + 64 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 64)^{2} \)
|
| $11$ |
\( T^{8} + 84T^{4} + 64 \)
|
| $13$ |
\( (T^{4} + 40 T^{2} + 128)^{2} \)
|
| $17$ |
\( (T^{2} + 8 T + 17)^{4} \)
|
| $19$ |
\( (T^{4} - 28 T^{2} + 128)^{2} \)
|
| $23$ |
\( (T^{4} + 14 T^{3} + \cdots + 256)^{2} \)
|
| $29$ |
\( T^{8} + 8448 T^{4} + 4194304 \)
|
| $31$ |
\( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 64)^{2} \)
|
| $37$ |
\( T^{8} + 5376 T^{4} + 262144 \)
|
| $41$ |
\( (T^{4} - 8 T^{3} + \cdots + 676)^{2} \)
|
| $43$ |
\( (T^{4} - 40 T^{2} + 128)^{2} \)
|
| $47$ |
\( (T^{2} - 14 T + 32)^{4} \)
|
| $53$ |
\( (T^{4} - 80 T^{2} + 512)^{2} \)
|
| $59$ |
\( (T^{4} - 40 T^{2} + 128)^{2} \)
|
| $61$ |
\( T^{8} + 21504 T^{4} + 4194304 \)
|
| $67$ |
\( (T^{4} + 40 T^{2} + 128)^{2} \)
|
| $71$ |
\( (T^{4} + 22 T^{3} + \cdots + 256)^{2} \)
|
| $73$ |
\( (T^{4} + 8 T^{3} + \cdots + 676)^{2} \)
|
| $79$ |
\( (T^{4} + 10 T^{3} + \cdots + 4096)^{2} \)
|
| $83$ |
\( (T^{4} - 296 T^{2} + 21632)^{2} \)
|
| $89$ |
\( (T^{2} - 2 T - 16)^{4} \)
|
| $97$ |
\( (T^{4} - 24 T^{3} + \cdots + 1444)^{2} \)
|
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