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: | \(4\) |
| Coefficient field: | 4.4.1506848.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 17x^{2} + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 153) |
| 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
\( x^{4} - 17x^{2} + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 11\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 34\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 11\beta_{2} + 34\beta_1 ) / 4 \)
|
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 | −11.3381 | 0 | 28.3925 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 5.10214 | 0 | 6.75787 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 7.60718 | 0 | −16.3925 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 20.6288 | 0 | 5.24213 | 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.bs | 4 | |
| 3.b | odd | 2 | 1 | 2448.4.a.bo | 4 | ||
| 4.b | odd | 2 | 1 | 153.4.a.i | yes | 4 | |
| 12.b | even | 2 | 1 | 153.4.a.h | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 153.4.a.h | ✓ | 4 | 12.b | even | 2 | 1 | |
| 153.4.a.i | yes | 4 | 4.b | odd | 2 | 1 | |
| 2448.4.a.bo | 4 | 3.b | odd | 2 | 1 | ||
| 2448.4.a.bs | 4 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2448))\):
|
\( T_{5}^{4} - 22T_{5}^{3} - 77T_{5}^{2} + 2612T_{5} - 9078 \)
|
|
\( T_{7}^{4} - 24T_{7}^{3} - 286T_{7}^{2} + 5160T_{7} - 16488 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 22 T^{3} + \cdots - 9078 \)
|
| $7$ |
\( T^{4} - 24 T^{3} + \cdots - 16488 \)
|
| $11$ |
\( T^{4} + 50 T^{3} + \cdots - 669528 \)
|
| $13$ |
\( T^{4} - 26 T^{3} + \cdots + 1030444 \)
|
| $17$ |
\( (T + 17)^{4} \)
|
| $19$ |
\( T^{4} + 34 T^{3} + \cdots + 4368204 \)
|
| $23$ |
\( T^{4} + 382 T^{3} + \cdots + 49041294 \)
|
| $29$ |
\( T^{4} - 540 T^{3} + \cdots - 232426584 \)
|
| $31$ |
\( T^{4} + \cdots - 2282423072 \)
|
| $37$ |
\( T^{4} + \cdots - 6443365976 \)
|
| $41$ |
\( T^{4} + \cdots + 9365221188 \)
|
| $43$ |
\( T^{4} + \cdots - 6091569648 \)
|
| $47$ |
\( T^{4} + 496 T^{3} + \cdots - 153753984 \)
|
| $53$ |
\( T^{4} + \cdots - 2914187328 \)
|
| $59$ |
\( T^{4} + \cdots + 23071976496 \)
|
| $61$ |
\( T^{4} + 1036 T^{3} + \cdots + 531291272 \)
|
| $67$ |
\( T^{4} + \cdots + 1798725056 \)
|
| $71$ |
\( T^{4} + \cdots + 82941464832 \)
|
| $73$ |
\( T^{4} + \cdots - 188519703168 \)
|
| $79$ |
\( T^{4} + \cdots - 23511973896 \)
|
| $83$ |
\( T^{4} + \cdots - 13135194432 \)
|
| $89$ |
\( T^{4} + \cdots + 113358314448 \)
|
| $97$ |
\( T^{4} + \cdots + 896966722688 \)
|
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