Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [656,6,Mod(1,656)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(656, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("656.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 656 = 2^{4} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 656.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: | \(105.211785797\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 492x^{2} - 5166x - 12280 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 82) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 492x^{2} - 5166x - 12280 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 27\nu^{2} + 88\nu - 2800 ) / 65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} - 43\nu^{2} - 1522\nu - 4920 ) / 65 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 18\beta _1 + 248 \)
|
| \(\nu^{3}\) | \(=\) |
\( 27\beta_{3} + 43\beta_{2} + 574\beta _1 + 3896 \)
|
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 | −26.1054 | 0 | −40.7661 | 0 | −102.718 | 0 | 438.491 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.35476 | 0 | 97.4041 | 0 | −219.239 | 0 | −155.488 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 11.4976 | 0 | 45.3504 | 0 | 37.0896 | 0 | −110.805 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 15.9626 | 0 | −89.9884 | 0 | 126.868 | 0 | 11.8031 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(41\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 656.6.a.b | 4 | |
| 4.b | odd | 2 | 1 | 82.6.a.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 738.6.a.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 82.6.a.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 656.6.a.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 738.6.a.h | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 8T_{3}^{3} - 546T_{3}^{2} - 198T_{3} + 44820 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(656))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 8 T^{3} + \cdots + 44820 \)
|
| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 16204824 \)
|
| $7$ |
\( T^{4} + 158 T^{3} + \cdots + 105966628 \)
|
| $11$ |
\( T^{4} + 2 T^{3} + \cdots - 337366620 \)
|
| $13$ |
\( T^{4} + 900 T^{3} + \cdots + 143706672 \)
|
| $17$ |
\( T^{4} - 1660 T^{3} + \cdots + 923839344 \)
|
| $19$ |
\( T^{4} + \cdots - 3740511005500 \)
|
| $23$ |
\( T^{4} + \cdots - 69689610155520 \)
|
| $29$ |
\( T^{4} + \cdots + 7482482178480 \)
|
| $31$ |
\( T^{4} + \cdots + 80754677049600 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!20 \)
|
| $41$ |
\( (T - 1681)^{4} \)
|
| $43$ |
\( T^{4} + \cdots + 61151127709568 \)
|
| $47$ |
\( T^{4} + \cdots - 87\!\cdots\!88 \)
|
| $53$ |
\( T^{4} + \cdots - 640391879919600 \)
|
| $59$ |
\( T^{4} + \cdots - 52\!\cdots\!80 \)
|
| $61$ |
\( T^{4} + \cdots + 26\!\cdots\!60 \)
|
| $67$ |
\( T^{4} + \cdots + 19\!\cdots\!40 \)
|
| $71$ |
\( T^{4} + \cdots - 67\!\cdots\!80 \)
|
| $73$ |
\( T^{4} + \cdots - 26\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots - 52\!\cdots\!60 \)
|
| $89$ |
\( T^{4} + \cdots + 96\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots - 21\!\cdots\!40 \)
|
| show more | |
| show less |