Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2496,4,Mod(1,2496)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, 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("2496.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2496 = 2^{6} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2496.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: | \(147.268767374\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 52x^{3} + 174x^{2} + 55x - 56 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 1248) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 52x^{3} + 174x^{2} + 55x - 56 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 43\nu^{2} + 61\nu + 15 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{3} + 8\nu^{2} - 109\nu - 7 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + \nu^{3} + 55\nu^{2} - 127\nu - 168 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{4} - 2\beta_{3} + 3\beta_{2} - \beta _1 + 87 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -8\beta_{4} + 12\beta_{3} - 8\beta_{2} + 39\beta _1 - 150 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 137\beta_{4} - 98\beta_{3} + 157\beta_{2} - 143\beta _1 + 3709 ) / 4 \)
|
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 | −3.00000 | 0 | −20.7291 | 0 | −25.3838 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −1.65766 | 0 | 21.2320 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −1.06282 | 0 | −7.20907 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | 11.7316 | 0 | −30.3794 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −3.00000 | 0 | 19.7180 | 0 | 3.74023 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(13\) | \( +1 \) |
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 | 2496.4.a.bz | 5 | |
| 4.b | odd | 2 | 1 | 2496.4.a.ce | 5 | ||
| 8.b | even | 2 | 1 | 1248.4.a.m | yes | 5 | |
| 8.d | odd | 2 | 1 | 1248.4.a.h | ✓ | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1248.4.a.h | ✓ | 5 | 8.d | odd | 2 | 1 | |
| 1248.4.a.m | yes | 5 | 8.b | even | 2 | 1 | |
| 2496.4.a.bz | 5 | 1.a | even | 1 | 1 | trivial | |
| 2496.4.a.ce | 5 | 4.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2496))\):
|
\( T_{5}^{5} - 8T_{5}^{4} - 448T_{5}^{3} + 3632T_{5}^{2} + 12304T_{5} + 8448 \)
|
|
\( T_{7}^{5} + 38T_{7}^{4} - 320T_{7}^{3} - 18736T_{7}^{2} - 45664T_{7} + 441472 \)
|
|
\( T_{11}^{5} - 12T_{11}^{4} - 3520T_{11}^{3} + 75552T_{11}^{2} + 158720T_{11} - 895488 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( (T + 3)^{5} \)
|
| $5$ |
\( T^{5} - 8 T^{4} + \cdots + 8448 \)
|
| $7$ |
\( T^{5} + 38 T^{4} + \cdots + 441472 \)
|
| $11$ |
\( T^{5} - 12 T^{4} + \cdots - 895488 \)
|
| $13$ |
\( (T + 13)^{5} \)
|
| $17$ |
\( T^{5} + \cdots + 2758020320 \)
|
| $19$ |
\( T^{5} - 82 T^{4} + \cdots + 250262656 \)
|
| $23$ |
\( T^{5} + \cdots - 6019743744 \)
|
| $29$ |
\( T^{5} + \cdots + 6128496480 \)
|
| $31$ |
\( T^{5} + \cdots + 379390098304 \)
|
| $37$ |
\( T^{5} + \cdots - 56288885472 \)
|
| $41$ |
\( T^{5} + \cdots + 438713495424 \)
|
| $43$ |
\( T^{5} + \cdots - 2381396900864 \)
|
| $47$ |
\( T^{5} + \cdots - 189609788416 \)
|
| $53$ |
\( T^{5} + \cdots + 538340590752 \)
|
| $59$ |
\( T^{5} + \cdots - 34929749269504 \)
|
| $61$ |
\( T^{5} + \cdots - 161541399797344 \)
|
| $67$ |
\( T^{5} + \cdots - 57941029666432 \)
|
| $71$ |
\( T^{5} + \cdots + 923885240320 \)
|
| $73$ |
\( T^{5} + \cdots - 509660519904 \)
|
| $79$ |
\( T^{5} + \cdots + 40420732247040 \)
|
| $83$ |
\( T^{5} + \cdots + 10\!\cdots\!16 \)
|
| $89$ |
\( T^{5} + \cdots + 16941742100928 \)
|
| $97$ |
\( T^{5} + \cdots - 708887603693280 \)
|
| show more | |
| show less |