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: | \(3\) |
| Coefficient field: | 3.3.3576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 15x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 189) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 15x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} - 11 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 6\nu + 9 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta _1 + 11 \)
|
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 |
|
−3.80992 | 0 | 6.51546 | −16.7551 | 0 | 0 | 5.65596 | 0 | 63.8356 | |||||||||||||||||||||||||||
| 1.2 | −0.670500 | 0 | −7.55043 | 9.86843 | 0 | 0 | 10.4266 | 0 | −6.61678 | ||||||||||||||||||||||||||||
| 1.3 | 5.48042 | 0 | 22.0350 | 4.88670 | 0 | 0 | 76.9175 | 0 | 26.7812 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 1323.4.a.z | 3 | |
| 3.b | odd | 2 | 1 | 1323.4.a.y | 3 | ||
| 7.b | odd | 2 | 1 | 189.4.a.k | yes | 3 | |
| 21.c | even | 2 | 1 | 189.4.a.j | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.4.a.j | ✓ | 3 | 21.c | even | 2 | 1 | |
| 189.4.a.k | yes | 3 | 7.b | odd | 2 | 1 | |
| 1323.4.a.y | 3 | 3.b | odd | 2 | 1 | ||
| 1323.4.a.z | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):
|
\( T_{2}^{3} - T_{2}^{2} - 22T_{2} - 14 \)
|
|
\( T_{5}^{3} + 2T_{5}^{2} - 199T_{5} + 808 \)
|
|
\( T_{13}^{3} + 21T_{13}^{2} - 1824T_{13} - 10592 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - T^{2} + \cdots - 14 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 2 T^{2} + \cdots + 808 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} - 116 T^{2} + \cdots - 35488 \)
|
| $13$ |
\( T^{3} + 21 T^{2} + \cdots - 10592 \)
|
| $17$ |
\( T^{3} - 63 T^{2} + \cdots + 555039 \)
|
| $19$ |
\( T^{3} - 66 T^{2} + \cdots + 59656 \)
|
| $23$ |
\( T^{3} - 159 T^{2} + \cdots + 154548 \)
|
| $29$ |
\( T^{3} - 173 T^{2} + \cdots + 10752188 \)
|
| $31$ |
\( T^{3} - 411 T^{2} + \cdots + 11998368 \)
|
| $37$ |
\( T^{3} - 540 T^{2} + \cdots - 2250234 \)
|
| $41$ |
\( T^{3} + 448 T^{2} + \cdots - 13925734 \)
|
| $43$ |
\( T^{3} + 99 T^{2} + \cdots - 8005583 \)
|
| $47$ |
\( T^{3} + 792 T^{2} + \cdots - 861678 \)
|
| $53$ |
\( T^{3} + 45 T^{2} + \cdots + 39848868 \)
|
| $59$ |
\( T^{3} + 207 T^{2} + \cdots - 3479247 \)
|
| $61$ |
\( T^{3} + 114 T^{2} + \cdots - 101463944 \)
|
| $67$ |
\( T^{3} + 669 T^{2} + \cdots - 4658224 \)
|
| $71$ |
\( T^{3} - 1029 T^{2} + \cdots + 183653568 \)
|
| $73$ |
\( T^{3} - 282 T^{2} + \cdots - 7203384 \)
|
| $79$ |
\( T^{3} - 684 T^{2} + \cdots - 65522918 \)
|
| $83$ |
\( T^{3} + 582 T^{2} + \cdots + 48300948 \)
|
| $89$ |
\( T^{3} + 71 T^{2} + \cdots - 437197844 \)
|
| $97$ |
\( T^{3} + 492 T^{2} + \cdots + 10780128 \)
|
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