Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2448,4,Mod(1,2448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2448.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2448.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: | \(144.436675694\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2636.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 14x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 17) |
| 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
\( x^{3} - 14x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 9 \)
|
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 | −3.03171 | 0 | 7.94049 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −0.885690 | 0 | −3.81828 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 11.9174 | 0 | −26.1222 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(17\) | \(-1\) |
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 | 2448.4.a.bi | 3 | |
| 3.b | odd | 2 | 1 | 272.4.a.h | 3 | ||
| 4.b | odd | 2 | 1 | 153.4.a.g | 3 | ||
| 12.b | even | 2 | 1 | 17.4.a.b | ✓ | 3 | |
| 24.f | even | 2 | 1 | 1088.4.a.v | 3 | ||
| 24.h | odd | 2 | 1 | 1088.4.a.x | 3 | ||
| 60.h | even | 2 | 1 | 425.4.a.g | 3 | ||
| 60.l | odd | 4 | 2 | 425.4.b.f | 6 | ||
| 84.h | odd | 2 | 1 | 833.4.a.d | 3 | ||
| 132.d | odd | 2 | 1 | 2057.4.a.e | 3 | ||
| 204.h | even | 2 | 1 | 289.4.a.b | 3 | ||
| 204.l | even | 4 | 2 | 289.4.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 17.4.a.b | ✓ | 3 | 12.b | even | 2 | 1 | |
| 153.4.a.g | 3 | 4.b | odd | 2 | 1 | ||
| 272.4.a.h | 3 | 3.b | odd | 2 | 1 | ||
| 289.4.a.b | 3 | 204.h | even | 2 | 1 | ||
| 289.4.b.b | 6 | 204.l | even | 4 | 2 | ||
| 425.4.a.g | 3 | 60.h | even | 2 | 1 | ||
| 425.4.b.f | 6 | 60.l | odd | 4 | 2 | ||
| 833.4.a.d | 3 | 84.h | odd | 2 | 1 | ||
| 1088.4.a.v | 3 | 24.f | even | 2 | 1 | ||
| 1088.4.a.x | 3 | 24.h | odd | 2 | 1 | ||
| 2057.4.a.e | 3 | 132.d | odd | 2 | 1 | ||
| 2448.4.a.bi | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2448))\):
|
\( T_{5}^{3} - 8T_{5}^{2} - 44T_{5} - 32 \)
|
|
\( T_{7}^{3} + 22T_{7}^{2} - 138T_{7} - 792 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots - 32 \)
|
| $7$ |
\( T^{3} + 22 T^{2} + \cdots - 792 \)
|
| $11$ |
\( T^{3} + 28 T^{2} + \cdots - 4692 \)
|
| $13$ |
\( T^{3} - 30 T^{2} + \cdots - 9392 \)
|
| $17$ |
\( (T - 17)^{3} \)
|
| $19$ |
\( T^{3} + 80 T^{2} + \cdots - 340128 \)
|
| $23$ |
\( T^{3} - 142 T^{2} + \cdots + 1600544 \)
|
| $29$ |
\( T^{3} - 456 T^{2} + \cdots - 1518624 \)
|
| $31$ |
\( T^{3} + 230 T^{2} + \cdots + 81608 \)
|
| $37$ |
\( T^{3} - 356 T^{2} + \cdots + 6176752 \)
|
| $41$ |
\( T^{3} - 294 T^{2} + \cdots + 1638744 \)
|
| $43$ |
\( T^{3} + 556 T^{2} + \cdots - 7270272 \)
|
| $47$ |
\( T^{3} - 640 T^{2} + \cdots - 1671168 \)
|
| $53$ |
\( T^{3} + 302 T^{2} + \cdots - 18162072 \)
|
| $59$ |
\( T^{3} - 636 T^{2} + \cdots + 49419072 \)
|
| $61$ |
\( T^{3} + 84 T^{2} + \cdots - 6792784 \)
|
| $67$ |
\( T^{3} + 1008 T^{2} + \cdots + 765952 \)
|
| $71$ |
\( T^{3} + 402 T^{2} + \cdots - 274866016 \)
|
| $73$ |
\( T^{3} - 838 T^{2} + \cdots - 19957512 \)
|
| $79$ |
\( T^{3} - 594 T^{2} + \cdots + 742135824 \)
|
| $83$ |
\( T^{3} + 2396 T^{2} + \cdots + 142080704 \)
|
| $89$ |
\( T^{3} - 170 T^{2} + \cdots + 446571376 \)
|
| $97$ |
\( T^{3} + 270 T^{2} + \cdots - 206623000 \)
|
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