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: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 324) |
| 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
\( x^{4} - 5x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 9\nu^{3} - 36\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -9\nu^{3} + 48\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 12\nu^{2} - 30 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 30 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{2} + 4\beta_1 ) / 9 \)
|
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 | −21.0707 | 0 | 31.4955 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −10.6784 | 0 | −23.4955 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 10.6784 | 0 | −23.4955 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 21.0707 | 0 | 31.4955 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
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 | 1296.4.a.z | 4 | |
| 3.b | odd | 2 | 1 | inner | 1296.4.a.z | 4 | |
| 4.b | odd | 2 | 1 | 324.4.a.e | ✓ | 4 | |
| 12.b | even | 2 | 1 | 324.4.a.e | ✓ | 4 | |
| 36.f | odd | 6 | 2 | 324.4.e.i | 8 | ||
| 36.h | even | 6 | 2 | 324.4.e.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.4.a.e | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 324.4.a.e | ✓ | 4 | 12.b | even | 2 | 1 | |
| 324.4.e.i | 8 | 36.f | odd | 6 | 2 | ||
| 324.4.e.i | 8 | 36.h | even | 6 | 2 | ||
| 1296.4.a.z | 4 | 1.a | even | 1 | 1 | trivial | |
| 1296.4.a.z | 4 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 558T_{5}^{2} + 50625 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1296))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 558 T^{2} + 50625 \)
|
| $7$ |
\( (T^{2} - 8 T - 740)^{2} \)
|
| $11$ |
\( T^{4} - 1368 T^{2} + 32400 \)
|
| $13$ |
\( (T^{2} - 58 T + 85)^{2} \)
|
| $17$ |
\( T^{4} - 11142 T^{2} + 12638025 \)
|
| $19$ |
\( (T^{2} + 112 T + 2380)^{2} \)
|
| $23$ |
\( T^{4} - 28152 T^{2} + 112784400 \)
|
| $29$ |
\( T^{4} - 64494 T^{2} + 386004609 \)
|
| $31$ |
\( (T^{2} + 184 T - 18752)^{2} \)
|
| $37$ |
\( (T^{2} - 334 T - 9155)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 1878702336 \)
|
| $43$ |
\( (T^{2} + 328 T - 34340)^{2} \)
|
| $47$ |
\( (T^{2} - 97200)^{2} \)
|
| $53$ |
\( T^{4} - 493632 T^{2} + 8294400 \)
|
| $59$ |
\( T^{4} + \cdots + 28482637824 \)
|
| $61$ |
\( (T^{2} - 502 T + 1765)^{2} \)
|
| $67$ |
\( (T^{2} + 160 T - 163700)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 18684702864 \)
|
| $73$ |
\( (T - 305)^{4} \)
|
| $79$ |
\( (T^{2} - 32 T - 472244)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 296284262400 \)
|
| $89$ |
\( T^{4} + \cdots + 3633221022201 \)
|
| $97$ |
\( (T^{2} - 1012 T + 243940)^{2} \)
|
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