Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1728,2,Mod(1,1728)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1728.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: | \(13.7981494693\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 54) |
| 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)
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 | 3.00000 | 0 | −1.00000 | 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 | 1728.2.a.y | 1 | |
| 3.b | odd | 2 | 1 | 1728.2.a.c | 1 | ||
| 4.b | odd | 2 | 1 | 1728.2.a.z | 1 | ||
| 8.b | even | 2 | 1 | 54.2.a.b | yes | 1 | |
| 8.d | odd | 2 | 1 | 432.2.a.b | 1 | ||
| 12.b | even | 2 | 1 | 1728.2.a.d | 1 | ||
| 24.f | even | 2 | 1 | 432.2.a.g | 1 | ||
| 24.h | odd | 2 | 1 | 54.2.a.a | ✓ | 1 | |
| 40.f | even | 2 | 1 | 1350.2.a.h | 1 | ||
| 40.i | odd | 4 | 2 | 1350.2.c.k | 2 | ||
| 56.h | odd | 2 | 1 | 2646.2.a.bd | 1 | ||
| 72.j | odd | 6 | 2 | 162.2.c.c | 2 | ||
| 72.l | even | 6 | 2 | 1296.2.i.c | 2 | ||
| 72.n | even | 6 | 2 | 162.2.c.b | 2 | ||
| 72.p | odd | 6 | 2 | 1296.2.i.o | 2 | ||
| 88.b | odd | 2 | 1 | 6534.2.a.b | 1 | ||
| 104.e | even | 2 | 1 | 9126.2.a.r | 1 | ||
| 120.i | odd | 2 | 1 | 1350.2.a.r | 1 | ||
| 120.w | even | 4 | 2 | 1350.2.c.b | 2 | ||
| 168.i | even | 2 | 1 | 2646.2.a.a | 1 | ||
| 264.m | even | 2 | 1 | 6534.2.a.bc | 1 | ||
| 312.b | odd | 2 | 1 | 9126.2.a.u | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 54.2.a.a | ✓ | 1 | 24.h | odd | 2 | 1 | |
| 54.2.a.b | yes | 1 | 8.b | even | 2 | 1 | |
| 162.2.c.b | 2 | 72.n | even | 6 | 2 | ||
| 162.2.c.c | 2 | 72.j | odd | 6 | 2 | ||
| 432.2.a.b | 1 | 8.d | odd | 2 | 1 | ||
| 432.2.a.g | 1 | 24.f | even | 2 | 1 | ||
| 1296.2.i.c | 2 | 72.l | even | 6 | 2 | ||
| 1296.2.i.o | 2 | 72.p | odd | 6 | 2 | ||
| 1350.2.a.h | 1 | 40.f | even | 2 | 1 | ||
| 1350.2.a.r | 1 | 120.i | odd | 2 | 1 | ||
| 1350.2.c.b | 2 | 120.w | even | 4 | 2 | ||
| 1350.2.c.k | 2 | 40.i | odd | 4 | 2 | ||
| 1728.2.a.c | 1 | 3.b | odd | 2 | 1 | ||
| 1728.2.a.d | 1 | 12.b | even | 2 | 1 | ||
| 1728.2.a.y | 1 | 1.a | even | 1 | 1 | trivial | |
| 1728.2.a.z | 1 | 4.b | odd | 2 | 1 | ||
| 2646.2.a.a | 1 | 168.i | even | 2 | 1 | ||
| 2646.2.a.bd | 1 | 56.h | odd | 2 | 1 | ||
| 6534.2.a.b | 1 | 88.b | odd | 2 | 1 | ||
| 6534.2.a.bc | 1 | 264.m | even | 2 | 1 | ||
| 9126.2.a.r | 1 | 104.e | even | 2 | 1 | ||
| 9126.2.a.u | 1 | 312.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1728))\):
|
\( T_{5} - 3 \)
|
|
\( T_{7} + 1 \)
|
|
\( T_{11} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T - 3 \)
|
| $7$ |
\( T + 1 \)
|
| $11$ |
\( T + 3 \)
|
| $13$ |
\( T - 4 \)
|
| $17$ |
\( T \)
|
| $19$ |
\( T + 2 \)
|
| $23$ |
\( T - 6 \)
|
| $29$ |
\( T - 6 \)
|
| $31$ |
\( T - 5 \)
|
| $37$ |
\( T + 2 \)
|
| $41$ |
\( T - 6 \)
|
| $43$ |
\( T - 10 \)
|
| $47$ |
\( T + 6 \)
|
| $53$ |
\( T - 9 \)
|
| $59$ |
\( T - 12 \)
|
| $61$ |
\( T + 8 \)
|
| $67$ |
\( T + 14 \)
|
| $71$ |
\( T \)
|
| $73$ |
\( T + 7 \)
|
| $79$ |
\( T - 8 \)
|
| $83$ |
\( T + 3 \)
|
| $89$ |
\( T - 18 \)
|
| $97$ |
\( T + 1 \)
|
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