Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1323,4,Mod(1,1323)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1323.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1323 = 3^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1323.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: | \(78.0595269376\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - 42x^{4} + 369x^{2} - 112 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{4} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 42x^{4} + 369x^{2} - 112 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 38\nu^{3} + 265\nu ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 32\nu^{2} + 106 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 44\nu^{3} - 409\nu ) / 6 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + 24\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{4} + 32\beta_{2} + 342 \)
|
| \(\nu^{5}\) | \(=\) |
\( 38\beta_{5} + 88\beta_{3} + 647\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.45019 | 0 | 21.7046 | −19.3432 | 0 | 0 | −74.6929 | 0 | 105.424 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.46131 | 0 | 3.98067 | 6.55599 | 0 | 0 | 13.9122 | 0 | −22.6923 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.560992 | 0 | −7.68529 | −12.9561 | 0 | 0 | 8.79932 | 0 | 7.26827 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.560992 | 0 | −7.68529 | 12.9561 | 0 | 0 | −8.79932 | 0 | 7.26827 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 3.46131 | 0 | 3.98067 | −6.55599 | 0 | 0 | −13.9122 | 0 | −22.6923 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.45019 | 0 | 21.7046 | 19.3432 | 0 | 0 | 74.6929 | 0 | 105.424 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1323.4.a.bg | yes | 6 |
| 3.b | odd | 2 | 1 | inner | 1323.4.a.bg | yes | 6 |
| 7.b | odd | 2 | 1 | 1323.4.a.bf | ✓ | 6 | |
| 21.c | even | 2 | 1 | 1323.4.a.bf | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1323.4.a.bf | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 1323.4.a.bf | ✓ | 6 | 21.c | even | 2 | 1 | |
| 1323.4.a.bg | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 1323.4.a.bg | yes | 6 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):
|
\( T_{2}^{6} - 42T_{2}^{4} + 369T_{2}^{2} - 112 \)
|
|
\( T_{5}^{6} - 585T_{5}^{4} + 86103T_{5}^{2} - 2699487 \)
|
|
\( T_{13}^{3} - 54T_{13}^{2} - 684T_{13} + 48168 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 42 T^{4} + \cdots - 112 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 585 T^{4} + \cdots - 2699487 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} - 2457 T^{4} + \cdots - 1539727 \)
|
| $13$ |
\( (T^{3} - 54 T^{2} + \cdots + 48168)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 143993106432 \)
|
| $19$ |
\( (T^{3} - 99 T^{2} + \cdots + 450387)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 16248512287 \)
|
| $29$ |
\( T^{6} + \cdots - 1629346543552 \)
|
| $31$ |
\( (T^{3} + 45 T^{2} + \cdots + 1675863)^{2} \)
|
| $37$ |
\( (T^{3} + 201 T^{2} + \cdots - 8763417)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 249467996315727 \)
|
| $43$ |
\( (T^{3} + 330 T^{2} + \cdots - 819288)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 135023220900288 \)
|
| $53$ |
\( T^{6} + \cdots - 38601195716608 \)
|
| $59$ |
\( T^{6} + \cdots - 11\!\cdots\!48 \)
|
| $61$ |
\( (T^{3} - 576 T^{2} + \cdots + 9597312)^{2} \)
|
| $67$ |
\( (T^{3} - 462 T^{2} + \cdots + 4527656)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 232630513987207 \)
|
| $73$ |
\( (T^{3} - 630 T^{2} + \cdots - 78997464)^{2} \)
|
| $79$ |
\( (T^{3} + 750 T^{2} + \cdots + 20170072)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 99\!\cdots\!12 \)
|
| $89$ |
\( T^{6} + \cdots - 12\!\cdots\!07 \)
|
| $97$ |
\( (T^{3} - 1656 T^{2} + \cdots + 913012992)^{2} \)
|
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