Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [648,4,Mod(1,648)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("648.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 648 = 2^{3} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 648.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: | \(38.2332376837\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.72153.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + 3x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 72) |
| 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
\( x^{4} - x^{3} - 8x^{2} + 3x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu^{3} - 3\nu^{2} - 12\nu + 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 5\nu^{2} - 17 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 5\nu^{2} + 6\nu - 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} + \beta_{2} + \beta _1 + 51 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{3} - 7\beta_{2} + 5\beta _1 + 51 ) / 12 \)
|
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 | 0 | 0 | −17.3189 | 0 | −2.37353 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1.69184 | 0 | −17.1413 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 5.99447 | 0 | 15.5776 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 8.01626 | 0 | 0.937231 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
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 | 648.4.a.h | 4 | |
| 3.b | odd | 2 | 1 | 648.4.a.i | 4 | ||
| 4.b | odd | 2 | 1 | 1296.4.a.y | 4 | ||
| 9.c | even | 3 | 2 | 216.4.i.a | 8 | ||
| 9.d | odd | 6 | 2 | 72.4.i.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 1296.4.a.ba | 4 | ||
| 36.f | odd | 6 | 2 | 432.4.i.e | 8 | ||
| 36.h | even | 6 | 2 | 144.4.i.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.4.i.a | ✓ | 8 | 9.d | odd | 6 | 2 | |
| 144.4.i.e | 8 | 36.h | even | 6 | 2 | ||
| 216.4.i.a | 8 | 9.c | even | 3 | 2 | ||
| 432.4.i.e | 8 | 36.f | odd | 6 | 2 | ||
| 648.4.a.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 648.4.a.i | 4 | 3.b | odd | 2 | 1 | ||
| 1296.4.a.y | 4 | 4.b | odd | 2 | 1 | ||
| 1296.4.a.ba | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 5T_{5}^{3} - 189T_{5}^{2} + 503T_{5} + 1408 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(648))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 5 T^{3} + \cdots + 1408 \)
|
| $7$ |
\( T^{4} + 3 T^{3} + \cdots + 594 \)
|
| $11$ |
\( T^{4} - 16 T^{3} + \cdots + 38929 \)
|
| $13$ |
\( T^{4} + 29 T^{3} + \cdots + 141262 \)
|
| $17$ |
\( T^{4} - 17 T^{3} + \cdots + 2567224 \)
|
| $19$ |
\( T^{4} + 109 T^{3} + \cdots - 5081408 \)
|
| $23$ |
\( T^{4} - 37 T^{3} + \cdots + 37951678 \)
|
| $29$ |
\( T^{4} + \cdots + 1572741522 \)
|
| $31$ |
\( T^{4} + \cdots - 3182697596 \)
|
| $37$ |
\( T^{4} + 366 T^{3} + \cdots - 215981856 \)
|
| $41$ |
\( T^{4} + 378 T^{3} + \cdots - 360819333 \)
|
| $43$ |
\( T^{4} + 506 T^{3} + \cdots - 567289217 \)
|
| $47$ |
\( T^{4} - 171 T^{3} + \cdots - 310694022 \)
|
| $53$ |
\( T^{4} + \cdots - 15963093536 \)
|
| $59$ |
\( T^{4} + \cdots + 1415131633 \)
|
| $61$ |
\( T^{4} + \cdots - 48115815704 \)
|
| $67$ |
\( T^{4} + \cdots - 137320199621 \)
|
| $71$ |
\( T^{4} + \cdots + 9762389248 \)
|
| $73$ |
\( T^{4} + \cdots - 88005243128 \)
|
| $79$ |
\( T^{4} + \cdots - 98670843044 \)
|
| $83$ |
\( T^{4} + \cdots + 67888747828 \)
|
| $89$ |
\( T^{4} + \cdots + 22003976592 \)
|
| $97$ |
\( T^{4} + \cdots - 818649939857 \)
|
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