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: | \(3\) |
| Coefficient field: | 3.3.3261.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 11x - 6 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 11x - 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( -2\nu^{2} + 6\nu + 13 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 2\nu - 15 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + \beta _1 + 32 ) / 4 \)
|
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 | −5.70308 | 0 | 5.31408 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | −1.31856 | 0 | 19.5875 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 17.0216 | 0 | −24.9016 | 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.br | 3 | |
| 4.b | odd | 2 | 1 | 2496.4.a.bn | 3 | ||
| 8.b | even | 2 | 1 | 1248.4.a.a | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 1248.4.a.b | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1248.4.a.a | ✓ | 3 | 8.b | even | 2 | 1 | |
| 1248.4.a.b | yes | 3 | 8.d | odd | 2 | 1 | |
| 2496.4.a.bn | 3 | 4.b | odd | 2 | 1 | ||
| 2496.4.a.br | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2496))\):
|
\( T_{5}^{3} - 10T_{5}^{2} - 112T_{5} - 128 \)
|
|
\( T_{7}^{3} - 516T_{7} + 2592 \)
|
|
\( T_{11}^{3} + 24T_{11}^{2} - 192T_{11} - 4544 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} - 10 T^{2} + \cdots - 128 \)
|
| $7$ |
\( T^{3} - 516T + 2592 \)
|
| $11$ |
\( T^{3} + 24 T^{2} + \cdots - 4544 \)
|
| $13$ |
\( (T + 13)^{3} \)
|
| $17$ |
\( T^{3} + 138 T^{2} + \cdots + 42552 \)
|
| $19$ |
\( T^{3} - 64 T^{2} + \cdots - 57344 \)
|
| $23$ |
\( T^{3} - 96 T^{2} + \cdots + 392704 \)
|
| $29$ |
\( T^{3} - 50 T^{2} + \cdots + 1928 \)
|
| $31$ |
\( T^{3} + 32 T^{2} + \cdots + 180736 \)
|
| $37$ |
\( T^{3} + 66 T^{2} + \cdots - 5424904 \)
|
| $41$ |
\( T^{3} - 30 T^{2} + \cdots + 3442176 \)
|
| $43$ |
\( T^{3} + 364 T^{2} + \cdots - 46922176 \)
|
| $47$ |
\( T^{3} - 540 T^{2} + \cdots + 11999232 \)
|
| $53$ |
\( T^{3} + 62 T^{2} + \cdots - 2659736 \)
|
| $59$ |
\( T^{3} + 336 T^{2} + \cdots - 98741952 \)
|
| $61$ |
\( T^{3} - 78 T^{2} + \cdots + 16020152 \)
|
| $67$ |
\( T^{3} + 968 T^{2} + \cdots + 31931584 \)
|
| $71$ |
\( T^{3} + 180 T^{2} + \cdots - 284495616 \)
|
| $73$ |
\( T^{3} + 1074 T^{2} + \cdots - 360130888 \)
|
| $79$ |
\( T^{3} + 788 T^{2} + \cdots - 620199488 \)
|
| $83$ |
\( T^{3} + 1416 T^{2} + \cdots - 334516544 \)
|
| $89$ |
\( T^{3} + 3050 T^{2} + \cdots + 719246752 \)
|
| $97$ |
\( T^{3} + 2874 T^{2} + \cdots + 331088216 \)
|
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