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: | \(2\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 136) |
| 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 \(\beta = 2\sqrt{3}\). 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 | −0.928203 | 0 | 7.60770 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 12.9282 | 0 | 28.3923 | 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.z | 2 | |
| 3.b | odd | 2 | 1 | 272.4.a.f | 2 | ||
| 4.b | odd | 2 | 1 | 1224.4.a.d | 2 | ||
| 12.b | even | 2 | 1 | 136.4.a.a | ✓ | 2 | |
| 24.f | even | 2 | 1 | 1088.4.a.n | 2 | ||
| 24.h | odd | 2 | 1 | 1088.4.a.p | 2 | ||
| 204.h | even | 2 | 1 | 2312.4.a.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 136.4.a.a | ✓ | 2 | 12.b | even | 2 | 1 | |
| 272.4.a.f | 2 | 3.b | odd | 2 | 1 | ||
| 1088.4.a.n | 2 | 24.f | even | 2 | 1 | ||
| 1088.4.a.p | 2 | 24.h | odd | 2 | 1 | ||
| 1224.4.a.d | 2 | 4.b | odd | 2 | 1 | ||
| 2312.4.a.b | 2 | 204.h | even | 2 | 1 | ||
| 2448.4.a.z | 2 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2448))\):
|
\( T_{5}^{2} - 12T_{5} - 12 \)
|
|
\( T_{7}^{2} - 36T_{7} + 216 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 12T - 12 \)
|
| $7$ |
\( T^{2} - 36T + 216 \)
|
| $11$ |
\( T^{2} + 20T - 1928 \)
|
| $13$ |
\( T^{2} + 84T + 36 \)
|
| $17$ |
\( (T + 17)^{2} \)
|
| $19$ |
\( T^{2} + 32T - 17072 \)
|
| $23$ |
\( T^{2} - 44T - 7016 \)
|
| $29$ |
\( T^{2} - 396T + 31092 \)
|
| $31$ |
\( T^{2} + 116T - 13064 \)
|
| $37$ |
\( T^{2} + 140T - 92300 \)
|
| $41$ |
\( T^{2} - 60T + 708 \)
|
| $43$ |
\( T^{2} + 640T + 98512 \)
|
| $47$ |
\( T^{2} + 496T + 18304 \)
|
| $53$ |
\( T^{2} + 236T + 4516 \)
|
| $59$ |
\( T^{2} - 576T - 5808 \)
|
| $61$ |
\( T^{2} + 348T - 225516 \)
|
| $67$ |
\( T^{2} + 1528 T + 580624 \)
|
| $71$ |
\( T^{2} + 876T - 81768 \)
|
| $73$ |
\( T^{2} + 380T + 12868 \)
|
| $79$ |
\( T^{2} + 172T - 351752 \)
|
| $83$ |
\( T^{2} + 1024 T - 350768 \)
|
| $89$ |
\( T^{2} - 844 T - 1144604 \)
|
| $97$ |
\( T^{2} - 36T - 37308 \)
|
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