Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1296,4,Mod(1,1296)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1296, 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("1296.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1296 = 2^{4} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1296.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: | \(76.4664753674\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 162) |
| 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 \(\beta = 3\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
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 | 0.803848 | 0 | 2.39230 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 11.1962 | 0 | −18.3923 | 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 | 1296.4.a.s | 2 | |
| 3.b | odd | 2 | 1 | 1296.4.a.j | 2 | ||
| 4.b | odd | 2 | 1 | 162.4.a.h | yes | 2 | |
| 12.b | even | 2 | 1 | 162.4.a.e | ✓ | 2 | |
| 36.f | odd | 6 | 2 | 162.4.c.i | 4 | ||
| 36.h | even | 6 | 2 | 162.4.c.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 162.4.a.e | ✓ | 2 | 12.b | even | 2 | 1 | |
| 162.4.a.h | yes | 2 | 4.b | odd | 2 | 1 | |
| 162.4.c.i | 4 | 36.f | odd | 6 | 2 | ||
| 162.4.c.j | 4 | 36.h | even | 6 | 2 | ||
| 1296.4.a.j | 2 | 3.b | odd | 2 | 1 | ||
| 1296.4.a.s | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 12T_{5} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 12T + 9 \)
|
| $7$ |
\( T^{2} + 16T - 44 \)
|
| $11$ |
\( T^{2} + 36T - 1404 \)
|
| $13$ |
\( T^{2} - 10T - 5267 \)
|
| $17$ |
\( T^{2} - 120T + 333 \)
|
| $19$ |
\( T^{2} - 8T - 13052 \)
|
| $23$ |
\( T^{2} + 180T + 7668 \)
|
| $29$ |
\( T^{2} - 324T + 24921 \)
|
| $31$ |
\( T^{2} - 248T - 19616 \)
|
| $37$ |
\( T^{2} + 434T + 46981 \)
|
| $41$ |
\( T^{2} - 192T - 43056 \)
|
| $43$ |
\( T^{2} - 608T + 91444 \)
|
| $47$ |
\( T^{2} - 384T + 1872 \)
|
| $53$ |
\( T^{2} + 408T - 114336 \)
|
| $59$ |
\( T^{2} - 1008 T + 226368 \)
|
| $61$ |
\( T^{2} - 742T + 128893 \)
|
| $67$ |
\( T^{2} - 104T - 511484 \)
|
| $71$ |
\( T^{2} - 1140 T + 323172 \)
|
| $73$ |
\( T^{2} - 850T - 68207 \)
|
| $79$ |
\( T^{2} - 440T - 19100 \)
|
| $83$ |
\( T^{2} + 264T - 79776 \)
|
| $89$ |
\( T^{2} - 768T - 66411 \)
|
| $97$ |
\( T^{2} + 764T + 61252 \)
|
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