Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,4,Mod(65,256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.65");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.e (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: | \(15.1044889615\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.224054542336.12 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 199x^{4} + 14641 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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
\( x^{8} + 199x^{4} + 14641 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 320\nu^{2} ) / 2541 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -10\nu^{7} + 121\nu^{5} - 659\nu^{3} + 66671\nu ) / 27951 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 20\nu^{7} - 242\nu^{5} + 1318\nu^{3} - 21538\nu ) / 27951 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -20\nu^{7} - 242\nu^{5} - 1318\nu^{3} - 21538\nu ) / 27951 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\nu^{7} + 121\nu^{5} + 7699\nu^{3} + 10769\nu ) / 27951 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -21\nu^{6} + 242\nu^{4} - 1638\nu^{2} + 24079 ) / 2541 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -21\nu^{6} - 242\nu^{4} - 1638\nu^{2} - 24079 ) / 2541 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 42\beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 20\beta_{5} + 21\beta_{4} - 11\beta_{3} ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -21\beta_{7} + 21\beta_{6} - 398 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -231\beta_{4} - 320\beta_{3} - 178\beta_{2} ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -80\beta_{7} - 80\beta_{6} - 819\beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -1318\beta_{5} - 4179\beta_{4} + 3520\beta_{3} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(255\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −6.75813 | − | 6.75813i | 0 | 10.5574 | − | 10.5574i | 0 | 16.3447i | 0 | 64.3446i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.2 | 0 | −2.51549 | − | 2.51549i | 0 | −2.55744 | + | 2.55744i | 0 | 2.20255i | 0 | − | 14.3446i | 0 | ||||||||||||||||||||||||||||||||||||||
| 65.3 | 0 | 2.51549 | + | 2.51549i | 0 | −2.55744 | + | 2.55744i | 0 | − | 2.20255i | 0 | − | 14.3446i | 0 | |||||||||||||||||||||||||||||||||||||
| 65.4 | 0 | 6.75813 | + | 6.75813i | 0 | 10.5574 | − | 10.5574i | 0 | − | 16.3447i | 0 | 64.3446i | 0 | ||||||||||||||||||||||||||||||||||||||
| 193.1 | 0 | −6.75813 | + | 6.75813i | 0 | 10.5574 | + | 10.5574i | 0 | − | 16.3447i | 0 | − | 64.3446i | 0 | |||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −2.51549 | + | 2.51549i | 0 | −2.55744 | − | 2.55744i | 0 | − | 2.20255i | 0 | 14.3446i | 0 | ||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 2.51549 | − | 2.51549i | 0 | −2.55744 | − | 2.55744i | 0 | 2.20255i | 0 | 14.3446i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 6.75813 | − | 6.75813i | 0 | 10.5574 | + | 10.5574i | 0 | 16.3447i | 0 | − | 64.3446i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 16.e | even | 4 | 1 | inner |
| 16.f | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.4.e.b | yes | 8 |
| 4.b | odd | 2 | 1 | inner | 256.4.e.b | yes | 8 |
| 8.b | even | 2 | 1 | 256.4.e.a | ✓ | 8 | |
| 8.d | odd | 2 | 1 | 256.4.e.a | ✓ | 8 | |
| 16.e | even | 4 | 1 | 256.4.e.a | ✓ | 8 | |
| 16.e | even | 4 | 1 | inner | 256.4.e.b | yes | 8 |
| 16.f | odd | 4 | 1 | 256.4.e.a | ✓ | 8 | |
| 16.f | odd | 4 | 1 | inner | 256.4.e.b | yes | 8 |
| 32.g | even | 8 | 1 | 1024.4.a.e | 4 | ||
| 32.g | even | 8 | 1 | 1024.4.a.j | 4 | ||
| 32.g | even | 8 | 2 | 1024.4.b.i | 8 | ||
| 32.h | odd | 8 | 1 | 1024.4.a.e | 4 | ||
| 32.h | odd | 8 | 1 | 1024.4.a.j | 4 | ||
| 32.h | odd | 8 | 2 | 1024.4.b.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 256.4.e.a | ✓ | 8 | 8.b | even | 2 | 1 | |
| 256.4.e.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 256.4.e.a | ✓ | 8 | 16.e | even | 4 | 1 | |
| 256.4.e.a | ✓ | 8 | 16.f | odd | 4 | 1 | |
| 256.4.e.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 256.4.e.b | yes | 8 | 4.b | odd | 2 | 1 | inner |
| 256.4.e.b | yes | 8 | 16.e | even | 4 | 1 | inner |
| 256.4.e.b | yes | 8 | 16.f | odd | 4 | 1 | inner |
| 1024.4.a.e | 4 | 32.g | even | 8 | 1 | ||
| 1024.4.a.e | 4 | 32.h | odd | 8 | 1 | ||
| 1024.4.a.j | 4 | 32.g | even | 8 | 1 | ||
| 1024.4.a.j | 4 | 32.h | odd | 8 | 1 | ||
| 1024.4.b.i | 8 | 32.g | even | 8 | 2 | ||
| 1024.4.b.i | 8 | 32.h | odd | 8 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(256, [\chi])\):
|
\( T_{3}^{8} + 8504T_{3}^{4} + 1336336 \)
|
|
\( T_{5}^{4} - 16T_{5}^{3} + 128T_{5}^{2} + 864T_{5} + 2916 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 8504 T^{4} + 1336336 \)
|
| $5$ |
\( (T^{4} - 16 T^{3} + \cdots + 2916)^{2} \)
|
| $7$ |
\( (T^{4} + 272 T^{2} + 1296)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 19671318375696 \)
|
| $13$ |
\( (T^{4} - 88 T^{3} + \cdots + 777924)^{2} \)
|
| $17$ |
\( (T^{2} - 96 T + 756)^{4} \)
|
| $19$ |
\( T^{8} + \cdots + 7198725105936 \)
|
| $23$ |
\( (T^{4} + 9776 T^{2} + 22240656)^{2} \)
|
| $29$ |
\( (T^{4} - 464 T^{3} + \cdots + 515199204)^{2} \)
|
| $31$ |
\( (T^{4} - 105536 T^{2} + 76317696)^{2} \)
|
| $37$ |
\( (T^{4} + 184 T^{3} + \cdots + 19693631556)^{2} \)
|
| $41$ |
\( (T^{4} + 52576 T^{2} + 620607744)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 45\!\cdots\!36 \)
|
| $47$ |
\( (T^{4} - 138944 T^{2} + 4274021376)^{2} \)
|
| $53$ |
\( (T^{4} - 440 T^{3} + \cdots + 453434436)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 56\!\cdots\!56 \)
|
| $61$ |
\( (T^{4} + 568 T^{3} + \cdots + 104343212484)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!96 \)
|
| $71$ |
\( (T^{4} + 456752 T^{2} + 31922254224)^{2} \)
|
| $73$ |
\( (T^{4} + 1375072 T^{2} + 351042030144)^{2} \)
|
| $79$ |
\( (T^{4} - 1392896 T^{2} + 88223256576)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 17\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + 160096 T^{2} + 1947986496)^{2} \)
|
| $97$ |
\( (T^{2} + 1120 T + 168948)^{4} \)
|
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