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: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{5}, \sqrt{13})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 5\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 11\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 3\nu^{2} - 14 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 14 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{2} + 11\beta_1 ) / 3 \)
|
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 |
|
−5.15688 | 0 | 18.5934 | 17.0220 | 0 | 0 | −54.6288 | 0 | −87.7802 | ||||||||||||||||||||||||||||||
| 1.2 | −1.55133 | 0 | −5.59339 | 9.81086 | 0 | 0 | 21.0878 | 0 | −15.2198 | |||||||||||||||||||||||||||||||
| 1.3 | 1.55133 | 0 | −5.59339 | −9.81086 | 0 | 0 | −21.0878 | 0 | −15.2198 | |||||||||||||||||||||||||||||||
| 1.4 | 5.15688 | 0 | 18.5934 | −17.0220 | 0 | 0 | 54.6288 | 0 | −87.7802 | |||||||||||||||||||||||||||||||
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.ba | 4 | |
| 3.b | odd | 2 | 1 | inner | 1323.4.a.ba | 4 | |
| 7.b | odd | 2 | 1 | 189.4.a.l | ✓ | 4 | |
| 21.c | even | 2 | 1 | 189.4.a.l | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.4.a.l | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 189.4.a.l | ✓ | 4 | 21.c | even | 2 | 1 | |
| 1323.4.a.ba | 4 | 1.a | even | 1 | 1 | trivial | |
| 1323.4.a.ba | 4 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):
|
\( T_{2}^{4} - 29T_{2}^{2} + 64 \)
|
|
\( T_{5}^{4} - 386T_{5}^{2} + 27889 \)
|
|
\( T_{13}^{2} - 34T_{13} - 4976 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 29T^{2} + 64 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 386 T^{2} + 27889 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 5906 T^{2} + 4592449 \)
|
| $13$ |
\( (T^{2} - 34 T - 4976)^{2} \)
|
| $17$ |
\( (T^{2} - 180)^{2} \)
|
| $19$ |
\( (T^{2} + 86 T - 491)^{2} \)
|
| $23$ |
\( T^{4} - 40986 T^{2} + 363016809 \)
|
| $29$ |
\( T^{4} - 18404 T^{2} + 2849344 \)
|
| $31$ |
\( (T^{2} + 204 T - 4221)^{2} \)
|
| $37$ |
\( (T^{2} - 660 T + 103635)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 15513948025 \)
|
| $43$ |
\( (T^{2} - 58 T + 256)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 8950673664 \)
|
| $53$ |
\( T^{4} + \cdots + 43477254144 \)
|
| $59$ |
\( T^{4} + \cdots + 8087045184 \)
|
| $61$ |
\( (T^{2} - 1336 T + 436864)^{2} \)
|
| $67$ |
\( (T^{2} - 722 T - 80864)^{2} \)
|
| $71$ |
\( T^{4} - 848826 T^{2} + 68112009 \)
|
| $73$ |
\( (T^{2} - 198 T - 4824)^{2} \)
|
| $79$ |
\( (T^{2} - 610 T - 5840)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 43147598400 \)
|
| $89$ |
\( T^{4} + \cdots + 15083032969 \)
|
| $97$ |
\( (T^{2} - 312 T + 14976)^{2} \)
|
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