Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [256,8,Mod(1,256)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("256.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 256 = 2^{8} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 256.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: | \(79.9705665239\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 163x^{4} + 4820x^{2} - 15296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{27} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 163x^{4} + 4820x^{2} - 15296 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 171\nu^{3} - 5468\nu ) / 120 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -37\nu^{5} + 5367\nu^{3} - 101516\nu ) / 600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -77\nu^{5} + 12207\nu^{3} - 13036\nu ) / 600 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 8\nu^{4} - 728\nu^{2} - 5581 ) / 25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -104\nu^{4} + 15864\nu^{2} - 275047 ) / 75 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - 8\beta_1 ) / 512 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + 13\beta_{4} + 13904 ) / 256 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 105\beta_{3} - 425\beta_{2} + 1528\beta_1 ) / 512 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 273\beta_{5} + 1983\beta_{4} + 1443856 ) / 256 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 12487\beta_{3} - 67207\beta_{2} + 243592\beta_1 ) / 512 \)
|
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 | −76.9497 | 0 | 338.443 | 0 | −438.996 | 0 | 3734.25 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −40.2163 | 0 | −324.492 | 0 | −956.960 | 0 | −569.651 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −21.9408 | 0 | −184.916 | 0 | 1051.96 | 0 | −1705.60 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 21.9408 | 0 | 184.916 | 0 | 1051.96 | 0 | −1705.60 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 40.2163 | 0 | 324.492 | 0 | −956.960 | 0 | −569.651 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 76.9497 | 0 | −338.443 | 0 | −438.996 | 0 | 3734.25 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 256.8.a.q | 6 | |
| 4.b | odd | 2 | 1 | 256.8.a.r | 6 | ||
| 8.b | even | 2 | 1 | inner | 256.8.a.q | 6 | |
| 8.d | odd | 2 | 1 | 256.8.a.r | 6 | ||
| 16.e | even | 4 | 2 | 32.8.b.a | 6 | ||
| 16.f | odd | 4 | 2 | 8.8.b.a | ✓ | 6 | |
| 48.i | odd | 4 | 2 | 288.8.d.b | 6 | ||
| 48.k | even | 4 | 2 | 72.8.d.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.8.b.a | ✓ | 6 | 16.f | odd | 4 | 2 | |
| 32.8.b.a | 6 | 16.e | even | 4 | 2 | ||
| 72.8.d.b | 6 | 48.k | even | 4 | 2 | ||
| 256.8.a.q | 6 | 1.a | even | 1 | 1 | trivial | |
| 256.8.a.q | 6 | 8.b | even | 2 | 1 | inner | |
| 256.8.a.r | 6 | 4.b | odd | 2 | 1 | ||
| 256.8.a.r | 6 | 8.d | odd | 2 | 1 | ||
| 288.8.d.b | 6 | 48.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(256))\):
|
\( T_{3}^{6} - 8020T_{3}^{4} + 13205808T_{3}^{2} - 4610229696 \)
|
|
\( T_{5}^{6} - 254032T_{5}^{4} + 19577926400T_{5}^{2} - 412405245440000 \)
|
|
\( T_{7}^{3} + 344T_{7}^{2} - 1048384T_{7} - 441929216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots - 4610229696 \)
|
| $5$ |
\( T^{6} + \cdots - 412405245440000 \)
|
| $7$ |
\( (T^{3} + 344 T^{2} + \cdots - 441929216)^{2} \)
|
| $11$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots - 24\!\cdots\!00 \)
|
| $17$ |
\( (T^{3} + \cdots - 9112197964104)^{2} \)
|
| $19$ |
\( T^{6} + \cdots - 47\!\cdots\!04 \)
|
| $23$ |
\( (T^{3} + \cdots + 2134822184448)^{2} \)
|
| $29$ |
\( T^{6} + \cdots - 42\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 18\!\cdots\!28)^{2} \)
|
| $37$ |
\( T^{6} + \cdots - 61\!\cdots\!64 \)
|
| $41$ |
\( (T^{3} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{6} + \cdots - 77\!\cdots\!56 \)
|
| $47$ |
\( (T^{3} + \cdots - 15\!\cdots\!96)^{2} \)
|
| $53$ |
\( T^{6} + \cdots - 12\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots - 55\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots - 10\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 75\!\cdots\!24 \)
|
| $71$ |
\( (T^{3} + \cdots + 38\!\cdots\!92)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots - 21\!\cdots\!72)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 49\!\cdots\!40)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 63\!\cdots\!16 \)
|
| $89$ |
\( (T^{3} + \cdots - 11\!\cdots\!20)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 16\!\cdots\!24)^{2} \)
|
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