Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [255,2,Mod(169,255)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(255, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("255.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 255 = 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 255.d (of order \(2\), degree \(1\), 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: | \(2.03618525154\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.75200995984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 10x^{6} + 30x^{4} + 27x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 10x^{6} + 30x^{4} + 27x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 5\nu^{2} + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 8\nu^{5} - 16\nu^{3} - 5\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 10\nu^{5} - 28\nu^{3} - 17\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{6} + 10\nu^{5} + 16\nu^{4} + 30\nu^{3} + 32\nu^{2} + 27\nu + 10 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 10\nu^{5} + 16\nu^{4} - 30\nu^{3} + 32\nu^{2} - 27\nu + 10 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{5} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 5\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 6\beta_{7} - 6\beta_{6} - 7\beta_{5} + \beta_{4} + 24\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 8\beta_{3} + 24\beta_{2} - 61 \)
|
| \(\nu^{7}\) | \(=\) |
\( -32\beta_{7} + 32\beta_{6} + 40\beta_{5} - 10\beta_{4} - 117\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/255\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(86\) | \(241\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
− | 2.28106i | −1.00000 | −3.20326 | −1.26671 | + | 1.84267i | 2.28106i | −3.05760 | 2.74471i | 1.00000 | 4.20326 | + | 2.88945i | |||||||||||||||||||||||||||||||||||||
| 169.2 | − | 1.83507i | −1.00000 | −1.36747 | 1.82635 | + | 1.29013i | 1.83507i | 3.49749 | − | 1.16073i | 1.00000 | 2.36747 | − | 3.35148i | |||||||||||||||||||||||||||||||||||||
| 169.3 | − | 1.11627i | −1.00000 | 0.753937 | −2.22518 | + | 0.220433i | 1.11627i | 2.67764 | − | 3.07414i | 1.00000 | 0.246063 | + | 2.48390i | |||||||||||||||||||||||||||||||||||||
| 169.4 | − | 0.428026i | −1.00000 | 1.81679 | 1.16553 | − | 1.90828i | 0.428026i | −1.11753 | − | 1.63369i | 1.00000 | −0.816794 | − | 0.498879i | |||||||||||||||||||||||||||||||||||||
| 169.5 | 0.428026i | −1.00000 | 1.81679 | 1.16553 | + | 1.90828i | − | 0.428026i | −1.11753 | 1.63369i | 1.00000 | −0.816794 | + | 0.498879i | ||||||||||||||||||||||||||||||||||||||
| 169.6 | 1.11627i | −1.00000 | 0.753937 | −2.22518 | − | 0.220433i | − | 1.11627i | 2.67764 | 3.07414i | 1.00000 | 0.246063 | − | 2.48390i | ||||||||||||||||||||||||||||||||||||||
| 169.7 | 1.83507i | −1.00000 | −1.36747 | 1.82635 | − | 1.29013i | − | 1.83507i | 3.49749 | 1.16073i | 1.00000 | 2.36747 | + | 3.35148i | ||||||||||||||||||||||||||||||||||||||
| 169.8 | 2.28106i | −1.00000 | −3.20326 | −1.26671 | − | 1.84267i | − | 2.28106i | −3.05760 | − | 2.74471i | 1.00000 | 4.20326 | − | 2.88945i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 85.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 255.2.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 765.2.d.e | 8 | ||
| 5.b | even | 2 | 1 | 255.2.d.b | yes | 8 | |
| 5.c | odd | 4 | 2 | 1275.2.g.g | 16 | ||
| 15.d | odd | 2 | 1 | 765.2.d.d | 8 | ||
| 17.b | even | 2 | 1 | 255.2.d.b | yes | 8 | |
| 51.c | odd | 2 | 1 | 765.2.d.d | 8 | ||
| 85.c | even | 2 | 1 | inner | 255.2.d.a | ✓ | 8 |
| 85.g | odd | 4 | 2 | 1275.2.g.g | 16 | ||
| 255.h | odd | 2 | 1 | 765.2.d.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 255.2.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 255.2.d.a | ✓ | 8 | 85.c | even | 2 | 1 | inner |
| 255.2.d.b | yes | 8 | 5.b | even | 2 | 1 | |
| 255.2.d.b | yes | 8 | 17.b | even | 2 | 1 | |
| 765.2.d.d | 8 | 15.d | odd | 2 | 1 | ||
| 765.2.d.d | 8 | 51.c | odd | 2 | 1 | ||
| 765.2.d.e | 8 | 3.b | odd | 2 | 1 | ||
| 765.2.d.e | 8 | 255.h | odd | 2 | 1 | ||
| 1275.2.g.g | 16 | 5.c | odd | 4 | 2 | ||
| 1275.2.g.g | 16 | 85.g | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 2T_{7}^{3} - 13T_{7}^{2} + 18T_{7} + 32 \)
acting on \(S_{2}^{\mathrm{new}}(255, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 10 T^{6} + \cdots + 4 \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} + T^{7} + \cdots + 625 \)
|
| $7$ |
\( (T^{4} - 2 T^{3} - 13 T^{2} + \cdots + 32)^{2} \)
|
| $11$ |
\( T^{8} + 37 T^{6} + \cdots + 4 \)
|
| $13$ |
\( T^{8} + 63 T^{6} + \cdots + 2304 \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots + 83521 \)
|
| $19$ |
\( (T^{4} + 7 T^{3} + \cdots - 116)^{2} \)
|
| $23$ |
\( (T^{4} + 3 T^{3} + \cdots + 192)^{2} \)
|
| $29$ |
\( T^{8} + 38 T^{6} + \cdots + 1024 \)
|
| $31$ |
\( T^{8} + 84 T^{6} + \cdots + 2304 \)
|
| $37$ |
\( (T^{4} + 2 T^{3} - 55 T^{2} + \cdots - 76)^{2} \)
|
| $41$ |
\( T^{8} + 81 T^{6} + \cdots + 4096 \)
|
| $43$ |
\( T^{8} + 175 T^{6} + \cdots + 576 \)
|
| $47$ |
\( T^{8} + 174 T^{6} + \cdots + 839056 \)
|
| $53$ |
\( T^{8} + 162 T^{6} + \cdots + 652864 \)
|
| $59$ |
\( (T^{4} - 2 T^{3} + \cdots - 192)^{2} \)
|
| $61$ |
\( T^{8} + 200 T^{6} + \cdots + 9216 \)
|
| $67$ |
\( T^{8} + 468 T^{6} + \cdots + 91240704 \)
|
| $71$ |
\( T^{8} + 216 T^{6} + \cdots + 4096 \)
|
| $73$ |
\( (T^{4} - 6 T^{3} + \cdots + 1732)^{2} \)
|
| $79$ |
\( T^{8} + 516 T^{6} + \cdots + 148644864 \)
|
| $83$ |
\( T^{8} + 440 T^{6} + \cdots + 48664576 \)
|
| $89$ |
\( (T^{4} + 2 T^{3} + \cdots - 144)^{2} \)
|
| $97$ |
\( (T^{4} + 6 T^{3} + \cdots + 1312)^{2} \)
|
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