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: | \(4\) |
| Coefficient field: | 4.4.5225.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 8x^{2} + x + 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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
\( x^{4} - x^{3} - 8x^{2} + x + 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 4\nu + 3 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu^{2} + 2\nu - 11 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 4\beta_{2} + 6\beta _1 + 5 \)
|
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 | −2.04678 | 0 | 0.163765 | 0 | −3.50105 | 0 | 1.18929 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.919131 | 0 | 4.02435 | 0 | 3.72325 | 0 | −2.15520 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.91913 | 0 | −3.40632 | 0 | −0.869151 | 0 | 0.683063 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.04678 | 0 | −1.78180 | 0 | −0.353057 | 0 | 6.28284 | 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.dr | 4 | |
| 4.b | odd | 2 | 1 | 7744.2.a.di | 4 | ||
| 8.b | even | 2 | 1 | 1936.2.a.bb | 4 | ||
| 8.d | odd | 2 | 1 | 968.2.a.n | 4 | ||
| 11.b | odd | 2 | 1 | 7744.2.a.ds | 4 | ||
| 11.c | even | 5 | 2 | 704.2.m.i | 8 | ||
| 24.f | even | 2 | 1 | 8712.2.a.ce | 4 | ||
| 44.c | even | 2 | 1 | 7744.2.a.dh | 4 | ||
| 44.h | odd | 10 | 2 | 704.2.m.l | 8 | ||
| 88.b | odd | 2 | 1 | 1936.2.a.bc | 4 | ||
| 88.g | even | 2 | 1 | 968.2.a.m | 4 | ||
| 88.k | even | 10 | 2 | 968.2.i.p | 8 | ||
| 88.k | even | 10 | 2 | 968.2.i.t | 8 | ||
| 88.l | odd | 10 | 2 | 88.2.i.b | ✓ | 8 | |
| 88.l | odd | 10 | 2 | 968.2.i.s | 8 | ||
| 88.o | even | 10 | 2 | 176.2.m.d | 8 | ||
| 264.p | odd | 2 | 1 | 8712.2.a.cd | 4 | ||
| 264.w | even | 10 | 2 | 792.2.r.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.2.i.b | ✓ | 8 | 88.l | odd | 10 | 2 | |
| 176.2.m.d | 8 | 88.o | even | 10 | 2 | ||
| 704.2.m.i | 8 | 11.c | even | 5 | 2 | ||
| 704.2.m.l | 8 | 44.h | odd | 10 | 2 | ||
| 792.2.r.g | 8 | 264.w | even | 10 | 2 | ||
| 968.2.a.m | 4 | 88.g | even | 2 | 1 | ||
| 968.2.a.n | 4 | 8.d | odd | 2 | 1 | ||
| 968.2.i.p | 8 | 88.k | even | 10 | 2 | ||
| 968.2.i.s | 8 | 88.l | odd | 10 | 2 | ||
| 968.2.i.t | 8 | 88.k | even | 10 | 2 | ||
| 1936.2.a.bb | 4 | 8.b | even | 2 | 1 | ||
| 1936.2.a.bc | 4 | 88.b | odd | 2 | 1 | ||
| 7744.2.a.dh | 4 | 44.c | even | 2 | 1 | ||
| 7744.2.a.di | 4 | 4.b | odd | 2 | 1 | ||
| 7744.2.a.dr | 4 | 1.a | even | 1 | 1 | trivial | |
| 7744.2.a.ds | 4 | 11.b | odd | 2 | 1 | ||
| 8712.2.a.cd | 4 | 264.p | odd | 2 | 1 | ||
| 8712.2.a.ce | 4 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7744))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 7T_{3}^{2} + 8T_{3} + 11 \)
|
|
\( T_{5}^{4} + T_{5}^{3} - 15T_{5}^{2} - 22T_{5} + 4 \)
|
|
\( T_{7}^{4} + T_{7}^{3} - 13T_{7}^{2} - 16T_{7} - 4 \)
|
|
\( T_{13}^{4} + T_{13}^{3} - 23T_{13}^{2} - 66T_{13} - 44 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 2 T^{3} + \cdots + 11 \)
|
| $5$ |
\( T^{4} + T^{3} - 15 T^{2} + \cdots + 4 \)
|
| $7$ |
\( T^{4} + T^{3} - 13 T^{2} + \cdots - 4 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + T^{3} + \cdots - 44 \)
|
| $17$ |
\( T^{4} + 12 T^{3} + \cdots - 199 \)
|
| $19$ |
\( (T^{2} - 7 T + 11)^{2} \)
|
| $23$ |
\( T^{4} - 2 T^{3} + \cdots - 64 \)
|
| $29$ |
\( T^{4} - 9 T^{3} + \cdots - 44 \)
|
| $31$ |
\( T^{4} + 11 T^{3} + \cdots - 236 \)
|
| $37$ |
\( T^{4} + 13 T^{3} + \cdots - 44 \)
|
| $41$ |
\( T^{4} + 8 T^{3} + \cdots - 11 \)
|
| $43$ |
\( T^{4} - 3 T^{3} + \cdots - 16 \)
|
| $47$ |
\( T^{4} + 7 T^{3} + \cdots + 176 \)
|
| $53$ |
\( T^{4} + 11 T^{3} + \cdots - 176 \)
|
| $59$ |
\( (T^{2} - 5 T - 25)^{2} \)
|
| $61$ |
\( T^{4} - 17 T^{3} + \cdots - 1936 \)
|
| $67$ |
\( T^{4} + 5 T^{3} + \cdots - 176 \)
|
| $71$ |
\( T^{4} - 5 T^{3} + \cdots - 716 \)
|
| $73$ |
\( T^{4} + 6 T^{3} + \cdots + 781 \)
|
| $79$ |
\( T^{4} + 7 T^{3} + \cdots + 844 \)
|
| $83$ |
\( T^{4} - 20 T^{3} + \cdots - 11 \)
|
| $89$ |
\( T^{4} - 11 T^{3} + \cdots - 124 \)
|
| $97$ |
\( T^{4} - 4 T^{3} + \cdots - 491 \)
|
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