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: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 730x^{3} - 4674x^{2} + 68790x + 487116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 730x^{3} - 4674x^{2} + 68790x + 487116 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -293\nu^{4} + 4434\nu^{3} + 184079\nu^{2} - 1073385\nu - 19783785 ) / 352443 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -180\nu^{4} + 2323\nu^{3} + 94241\nu^{2} - 359901\nu - 5546457 ) / 117481 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -128\nu^{4} + 906\nu^{3} + 82307\nu^{2} - 828\nu - 6354186 ) / 50349 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 6\beta_{3} + 8\beta_{2} + 6\beta _1 + 292 \)
|
| \(\nu^{3}\) | \(=\) |
\( -47\beta_{4} - 11\beta_{3} + 164\beta_{2} + 465\beta _1 + 2755 \)
|
| \(\nu^{4}\) | \(=\) |
\( -83\beta_{4} - 3936\beta_{3} + 6305\beta_{2} + 7143\beta _1 + 157621 \)
|
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 | −17.2082 | 0 | −11.6574 | 0 | 121.255 | 0 | 53.1223 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −9.96111 | 0 | −33.0424 | 0 | −56.8900 | 0 | −143.776 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −5.27474 | 0 | 45.6181 | 0 | 237.714 | 0 | −215.177 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 12.3879 | 0 | −48.5965 | 0 | −247.165 | 0 | −89.5391 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 30.0561 | 0 | 9.67826 | 0 | −16.9135 | 0 | 660.370 | 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.c | 5 | |
| 4.b | odd | 2 | 1 | 82.6.a.c | ✓ | 5 | |
| 12.b | even | 2 | 1 | 738.6.a.k | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 82.6.a.c | ✓ | 5 | 4.b | odd | 2 | 1 | |
| 656.6.a.c | 5 | 1.a | even | 1 | 1 | trivial | |
| 738.6.a.k | 5 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - 10T_{3}^{4} - 690T_{3}^{3} - 374T_{3}^{2} + 78806T_{3} + 336648 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(656))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} - 10 T^{4} + \cdots + 336648 \)
|
| $5$ |
\( T^{5} + 38 T^{4} + \cdots + 8264472 \)
|
| $7$ |
\( T^{5} + \cdots + 6855089756 \)
|
| $11$ |
\( T^{5} + \cdots + 1903543296456 \)
|
| $13$ |
\( T^{5} + \cdots - 23069043640000 \)
|
| $17$ |
\( T^{5} + \cdots - 20\!\cdots\!88 \)
|
| $19$ |
\( T^{5} + \cdots - 645857447500368 \)
|
| $23$ |
\( T^{5} + \cdots + 61\!\cdots\!16 \)
|
| $29$ |
\( T^{5} + \cdots + 94\!\cdots\!44 \)
|
| $31$ |
\( T^{5} + \cdots - 59\!\cdots\!04 \)
|
| $37$ |
\( T^{5} + \cdots + 59\!\cdots\!32 \)
|
| $41$ |
\( (T + 1681)^{5} \)
|
| $43$ |
\( T^{5} + \cdots - 78\!\cdots\!88 \)
|
| $47$ |
\( T^{5} + \cdots + 82\!\cdots\!76 \)
|
| $53$ |
\( T^{5} + \cdots + 25\!\cdots\!92 \)
|
| $59$ |
\( T^{5} + \cdots + 29\!\cdots\!56 \)
|
| $61$ |
\( T^{5} + \cdots - 27\!\cdots\!24 \)
|
| $67$ |
\( T^{5} + \cdots - 13\!\cdots\!16 \)
|
| $71$ |
\( T^{5} + \cdots + 77\!\cdots\!68 \)
|
| $73$ |
\( T^{5} + \cdots - 10\!\cdots\!76 \)
|
| $79$ |
\( T^{5} + \cdots - 71\!\cdots\!12 \)
|
| $83$ |
\( T^{5} + \cdots + 63\!\cdots\!92 \)
|
| $89$ |
\( T^{5} + \cdots - 12\!\cdots\!64 \)
|
| $97$ |
\( T^{5} + \cdots + 15\!\cdots\!32 \)
|
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