Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1728,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1728.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1728 = 2^{6} \cdot 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(101.955300490\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.229.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 864) |
| 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
\( x^{3} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 6\nu^{2} - 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( -6\nu^{2} + 12\nu + 16 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta _1 + 16 ) / 6 \)
|
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 | −16.6126 | 0 | 11.5634 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 3.77559 | 0 | −28.1053 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 9.83700 | 0 | 13.5419 | 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 | 1728.4.a.by | 3 | |
| 3.b | odd | 2 | 1 | 1728.4.a.ca | 3 | ||
| 4.b | odd | 2 | 1 | 1728.4.a.bz | 3 | ||
| 8.b | even | 2 | 1 | 864.4.a.o | yes | 3 | |
| 8.d | odd | 2 | 1 | 864.4.a.p | yes | 3 | |
| 12.b | even | 2 | 1 | 1728.4.a.cb | 3 | ||
| 24.f | even | 2 | 1 | 864.4.a.n | yes | 3 | |
| 24.h | odd | 2 | 1 | 864.4.a.m | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 864.4.a.m | ✓ | 3 | 24.h | odd | 2 | 1 | |
| 864.4.a.n | yes | 3 | 24.f | even | 2 | 1 | |
| 864.4.a.o | yes | 3 | 8.b | even | 2 | 1 | |
| 864.4.a.p | yes | 3 | 8.d | odd | 2 | 1 | |
| 1728.4.a.by | 3 | 1.a | even | 1 | 1 | trivial | |
| 1728.4.a.bz | 3 | 4.b | odd | 2 | 1 | ||
| 1728.4.a.ca | 3 | 3.b | odd | 2 | 1 | ||
| 1728.4.a.cb | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1728))\):
|
\( T_{5}^{3} + 3T_{5}^{2} - 189T_{5} + 617 \)
|
|
\( T_{7}^{3} + 3T_{7}^{2} - 549T_{7} + 4401 \)
|
|
\( T_{11}^{3} + 21T_{11}^{2} - 1629T_{11} - 32697 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 3 T^{2} + \cdots + 617 \)
|
| $7$ |
\( T^{3} + 3 T^{2} + \cdots + 4401 \)
|
| $11$ |
\( T^{3} + 21 T^{2} + \cdots - 32697 \)
|
| $13$ |
\( T^{3} - 24 T^{2} + \cdots - 55296 \)
|
| $17$ |
\( T^{3} - 36 T^{2} + \cdots - 37888 \)
|
| $19$ |
\( T^{3} + 66 T^{2} + \cdots - 94824 \)
|
| $23$ |
\( T^{3} + 42 T^{2} + \cdots - 1415016 \)
|
| $29$ |
\( T^{3} - 90 T^{2} + \cdots - 194168 \)
|
| $31$ |
\( T^{3} - 351 T^{2} + \cdots + 7618779 \)
|
| $37$ |
\( T^{3} - 6 T^{2} + \cdots - 4834088 \)
|
| $41$ |
\( T^{3} + 138 T^{2} + \cdots + 138808 \)
|
| $43$ |
\( T^{3} + 498 T^{2} + \cdots - 18917928 \)
|
| $47$ |
\( T^{3} + 114 T^{2} + \cdots - 25598376 \)
|
| $53$ |
\( T^{3} + 345 T^{2} + \cdots + 1083259 \)
|
| $59$ |
\( T^{3} + 228 T^{2} + \cdots + 24864192 \)
|
| $61$ |
\( T^{3} + 168 T^{2} + \cdots - 152233984 \)
|
| $67$ |
\( T^{3} + 1506 T^{2} + \cdots + 73714968 \)
|
| $71$ |
\( T^{3} + 1020 T^{2} + \cdots - 74974464 \)
|
| $73$ |
\( T^{3} + 453 T^{2} + \cdots - 67223817 \)
|
| $79$ |
\( T^{3} - 2448 T^{2} + \cdots - 300837888 \)
|
| $83$ |
\( T^{3} + 957 T^{2} + \cdots - 250960977 \)
|
| $89$ |
\( T^{3} - 1038 T^{2} + \cdots + 423986584 \)
|
| $97$ |
\( T^{3} - 549 T^{2} + \cdots - 324025191 \)
|
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