Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [648,4,Mod(217,648)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("648.217");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 648 = 2^{3} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 648.i (of order \(3\), degree \(2\), not 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: | \(38.2332376837\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.897122304.10 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 8\nu^{6} - 51\nu^{4} + 408\nu^{2} - 169 ) / 663 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -32\nu^{7} + 13\nu^{6} + 204\nu^{5} - 1632\nu^{3} + 5980\nu + 1300 ) / 663 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 28\nu^{7} + 13\nu^{6} - 510\nu^{5} + 2754\nu^{3} - 9542\nu + 1300 ) / 663 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 100 ) / 17 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 32\nu^{7} + 32\nu^{6} - 204\nu^{5} - 204\nu^{4} + 1632\nu^{3} + 969\nu^{2} + 1976\nu - 676 ) / 663 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -19\nu^{6} + 204\nu^{4} - 969\nu^{2} + 1976 ) / 221 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -106\nu^{7} + 32\nu^{6} + 510\nu^{5} - 204\nu^{4} - 2754\nu^{3} + 969\nu^{2} - 2236\nu - 676 ) / 663 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 3\beta_{5} - 2\beta_{4} + 3\beta_{2} ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{4} + 12\beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 12\beta_{7} + 14\beta_{6} + 30\beta_{5} - 7\beta_{4} - 6\beta_{3} - 15\beta_{2} ) / 36 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 8\beta_{6} + 57\beta _1 - 57 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 48\beta_{7} + 43\beta_{6} + 81\beta_{5} + 43\beta_{4} - 96\beta_{3} - 162\beta_{2} ) / 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 17\beta_{4} - 100 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -306\beta_{7} - 253\beta_{6} - 453\beta_{5} + 506\beta_{4} - 306\beta_{3} - 453\beta_{2} ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(487\) | \(569\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 217.1 |
|
0 | 0 | 0 | −5.26031 | − | 9.11112i | 0 | 0.177444 | − | 0.307342i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||
| 217.2 | 0 | 0 | 0 | −2.90171 | − | 5.02591i | 0 | 13.0614 | − | 22.6230i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 217.3 | 0 | 0 | 0 | 3.16966 | + | 5.49001i | 0 | −9.59728 | + | 16.6230i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 217.4 | 0 | 0 | 0 | 8.99236 | + | 15.5752i | 0 | −3.64155 | + | 6.30734i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 433.1 | 0 | 0 | 0 | −5.26031 | + | 9.11112i | 0 | 0.177444 | + | 0.307342i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 433.2 | 0 | 0 | 0 | −2.90171 | + | 5.02591i | 0 | 13.0614 | + | 22.6230i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 433.3 | 0 | 0 | 0 | 3.16966 | − | 5.49001i | 0 | −9.59728 | − | 16.6230i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
| 433.4 | 0 | 0 | 0 | 8.99236 | − | 15.5752i | 0 | −3.64155 | − | 6.30734i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 648.4.i.v | 8 | |
| 3.b | odd | 2 | 1 | 648.4.i.u | 8 | ||
| 9.c | even | 3 | 1 | 648.4.a.g | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 648.4.i.v | 8 | |
| 9.d | odd | 6 | 1 | 648.4.a.j | yes | 4 | |
| 9.d | odd | 6 | 1 | 648.4.i.u | 8 | ||
| 36.f | odd | 6 | 1 | 1296.4.a.x | 4 | ||
| 36.h | even | 6 | 1 | 1296.4.a.bb | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 648.4.a.g | ✓ | 4 | 9.c | even | 3 | 1 | |
| 648.4.a.j | yes | 4 | 9.d | odd | 6 | 1 | |
| 648.4.i.u | 8 | 3.b | odd | 2 | 1 | ||
| 648.4.i.u | 8 | 9.d | odd | 6 | 1 | ||
| 648.4.i.v | 8 | 1.a | even | 1 | 1 | trivial | |
| 648.4.i.v | 8 | 9.c | even | 3 | 1 | inner | |
| 1296.4.a.x | 4 | 36.f | odd | 6 | 1 | ||
| 1296.4.a.bb | 4 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 8 T_{5}^{7} + 286 T_{5}^{6} + 1024 T_{5}^{5} + 45331 T_{5}^{4} + 27904 T_{5}^{3} + \cdots + 48455521 \)
acting on \(S_{4}^{\mathrm{new}}(648, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 8 T^{7} + \cdots + 48455521 \)
|
| $7$ |
\( T^{8} + 552 T^{6} + \cdots + 1679616 \)
|
| $11$ |
\( T^{8} + \cdots + 3729789184 \)
|
| $13$ |
\( T^{8} + \cdots + 1531200831889 \)
|
| $17$ |
\( (T^{4} + 16 T^{3} + \cdots + 14319721)^{2} \)
|
| $19$ |
\( (T^{4} - 80 T^{3} + \cdots + 31790992)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 23\!\cdots\!44 \)
|
| $29$ |
\( T^{8} + \cdots + 30\!\cdots\!89 \)
|
| $31$ |
\( T^{8} + \cdots + 23\!\cdots\!44 \)
|
| $37$ |
\( (T^{4} + 276 T^{3} + \cdots + 948334041)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 41\!\cdots\!64 \)
|
| $43$ |
\( T^{8} + \cdots + 26\!\cdots\!16 \)
|
| $47$ |
\( T^{8} + \cdots + 13\!\cdots\!96 \)
|
| $53$ |
\( (T^{4} + 944 T^{3} + \cdots - 22310316032)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 35\!\cdots\!96 \)
|
| $61$ |
\( T^{8} + \cdots + 10\!\cdots\!81 \)
|
| $67$ |
\( T^{8} + \cdots + 94\!\cdots\!84 \)
|
| $71$ |
\( (T^{4} + 1720 T^{3} + \cdots - 9807661424)^{2} \)
|
| $73$ |
\( (T^{4} + 764 T^{3} + \cdots - 194677777919)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots + 37\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots - 1312975087623)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 89\!\cdots\!84 \)
|
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