Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7744,2,Mod(1,7744)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7744, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7744.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7744 = 2^{6} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7744.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: | \(61.8361513253\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.404.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 5x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 3872) |
| 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
\( x^{3} - x^{2} - 5x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 4 \)
|
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 | −2.53407 | 0 | −1.53407 | 0 | 4.95558 | 0 | 3.42151 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.517304 | 0 | 0.482696 | 0 | −3.21509 | 0 | −2.73240 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.05137 | 0 | 4.05137 | 0 | 2.25951 | 0 | 6.31088 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
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 | 7744.2.a.df | 3 | |
| 4.b | odd | 2 | 1 | 7744.2.a.dd | 3 | ||
| 8.b | even | 2 | 1 | 3872.2.a.be | yes | 3 | |
| 8.d | odd | 2 | 1 | 3872.2.a.bc | yes | 3 | |
| 11.b | odd | 2 | 1 | 7744.2.a.de | 3 | ||
| 44.c | even | 2 | 1 | 7744.2.a.dg | 3 | ||
| 88.b | odd | 2 | 1 | 3872.2.a.bb | ✓ | 3 | |
| 88.g | even | 2 | 1 | 3872.2.a.bd | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3872.2.a.bb | ✓ | 3 | 88.b | odd | 2 | 1 | |
| 3872.2.a.bc | yes | 3 | 8.d | odd | 2 | 1 | |
| 3872.2.a.bd | yes | 3 | 88.g | even | 2 | 1 | |
| 3872.2.a.be | yes | 3 | 8.b | even | 2 | 1 | |
| 7744.2.a.dd | 3 | 4.b | odd | 2 | 1 | ||
| 7744.2.a.de | 3 | 11.b | odd | 2 | 1 | ||
| 7744.2.a.df | 3 | 1.a | even | 1 | 1 | trivial | |
| 7744.2.a.dg | 3 | 44.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):
|
\( T_{3}^{3} - 8T_{3} - 4 \)
|
|
\( T_{5}^{3} - 3T_{5}^{2} - 5T_{5} + 3 \)
|
|
\( T_{7}^{3} - 4T_{7}^{2} - 12T_{7} + 36 \)
|
|
\( T_{13}^{3} + 7T_{13}^{2} - T_{13} - 43 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 8T - 4 \)
|
| $5$ |
\( T^{3} - 3 T^{2} + \cdots + 3 \)
|
| $7$ |
\( T^{3} - 4 T^{2} + \cdots + 36 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + 7T^{2} - T - 43 \)
|
| $17$ |
\( T^{3} + T^{2} + \cdots + 43 \)
|
| $19$ |
\( T^{3} + 12 T^{2} + \cdots + 36 \)
|
| $23$ |
\( T^{3} + 4 T^{2} + \cdots - 36 \)
|
| $29$ |
\( T^{3} - T^{2} + \cdots - 43 \)
|
| $31$ |
\( T^{3} + 8 T^{2} + \cdots - 148 \)
|
| $37$ |
\( T^{3} + T^{2} + \cdots - 21 \)
|
| $41$ |
\( T^{3} - 3 T^{2} + \cdots + 63 \)
|
| $43$ |
\( T^{3} + 12 T^{2} + \cdots - 96 \)
|
| $47$ |
\( T^{3} - 8 T^{2} + \cdots + 12 \)
|
| $53$ |
\( T^{3} - 15 T^{2} + \cdots - 81 \)
|
| $59$ |
\( T^{3} + 16 T^{2} + \cdots - 96 \)
|
| $61$ |
\( (T - 6)^{3} \)
|
| $67$ |
\( T^{3} + 8 T^{2} + \cdots - 252 \)
|
| $71$ |
\( (T + 8)^{3} \)
|
| $73$ |
\( T^{3} + 14 T^{2} + \cdots - 648 \)
|
| $79$ |
\( T^{3} - 8 T^{2} + \cdots + 796 \)
|
| $83$ |
\( T^{3} + 4 T^{2} + \cdots - 1036 \)
|
| $89$ |
\( T^{3} - 25 T^{2} + \cdots - 379 \)
|
| $97$ |
\( T^{3} - 9 T^{2} + \cdots + 2189 \)
|
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