Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1216,2,Mod(1,1216)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1216, 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("1216.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1216 = 2^{6} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1216.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: | \(9.70980888579\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.961.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 10x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 152) |
| 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
\( x^{3} - x^{2} - 10x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} - \beta _1 + 8 \)
|
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 | −3.29707 | 0 | 3.08387 | 0 | 1.78680 | 0 | 7.87067 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.786802 | 0 | −3.29707 | 0 | −2.08387 | 0 | −2.38094 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.08387 | 0 | −0.786802 | 0 | 4.29707 | 0 | 6.51027 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(19\) | \( -1 \) |
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 | 1216.2.a.u | 3 | |
| 4.b | odd | 2 | 1 | 1216.2.a.v | 3 | ||
| 8.b | even | 2 | 1 | 152.2.a.c | ✓ | 3 | |
| 8.d | odd | 2 | 1 | 304.2.a.g | 3 | ||
| 24.f | even | 2 | 1 | 2736.2.a.bd | 3 | ||
| 24.h | odd | 2 | 1 | 1368.2.a.n | 3 | ||
| 40.e | odd | 2 | 1 | 7600.2.a.bv | 3 | ||
| 40.f | even | 2 | 1 | 3800.2.a.r | 3 | ||
| 40.i | odd | 4 | 2 | 3800.2.d.j | 6 | ||
| 56.h | odd | 2 | 1 | 7448.2.a.bf | 3 | ||
| 152.b | even | 2 | 1 | 5776.2.a.bp | 3 | ||
| 152.g | odd | 2 | 1 | 2888.2.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 152.2.a.c | ✓ | 3 | 8.b | even | 2 | 1 | |
| 304.2.a.g | 3 | 8.d | odd | 2 | 1 | ||
| 1216.2.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
| 1216.2.a.v | 3 | 4.b | odd | 2 | 1 | ||
| 1368.2.a.n | 3 | 24.h | odd | 2 | 1 | ||
| 2736.2.a.bd | 3 | 24.f | even | 2 | 1 | ||
| 2888.2.a.o | 3 | 152.g | odd | 2 | 1 | ||
| 3800.2.a.r | 3 | 40.f | even | 2 | 1 | ||
| 3800.2.d.j | 6 | 40.i | odd | 4 | 2 | ||
| 5776.2.a.bp | 3 | 152.b | even | 2 | 1 | ||
| 7448.2.a.bf | 3 | 56.h | odd | 2 | 1 | ||
| 7600.2.a.bv | 3 | 40.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1216))\):
|
\( T_{3}^{3} + T_{3}^{2} - 10T_{3} - 8 \)
|
|
\( T_{5}^{3} + T_{5}^{2} - 10T_{5} - 8 \)
|
|
\( T_{7}^{3} - 4T_{7}^{2} - 5T_{7} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + T^{2} - 10T - 8 \)
|
| $5$ |
\( T^{3} + T^{2} - 10T - 8 \)
|
| $7$ |
\( T^{3} - 4 T^{2} + \cdots + 16 \)
|
| $11$ |
\( T^{3} - 5 T^{2} + \cdots + 8 \)
|
| $13$ |
\( T^{3} + 5 T^{2} + \cdots - 8 \)
|
| $17$ |
\( T^{3} - 2 T^{2} + \cdots + 2 \)
|
| $19$ |
\( (T - 1)^{3} \)
|
| $23$ |
\( T^{3} + 5 T^{2} + \cdots - 256 \)
|
| $29$ |
\( T^{3} - 9 T^{2} + \cdots + 4 \)
|
| $31$ |
\( T^{3} \)
|
| $37$ |
\( (T - 2)^{3} \)
|
| $41$ |
\( T^{3} - 8 T^{2} + \cdots + 128 \)
|
| $43$ |
\( T^{3} + 17 T^{2} + \cdots - 368 \)
|
| $47$ |
\( T^{3} + T^{2} + \cdots - 256 \)
|
| $53$ |
\( T^{3} + T^{2} + \cdots - 256 \)
|
| $59$ |
\( T^{3} - 23 T^{2} + \cdots - 376 \)
|
| $61$ |
\( T^{3} + 3 T^{2} + \cdots - 92 \)
|
| $67$ |
\( T^{3} + 15 T^{2} + \cdots + 32 \)
|
| $71$ |
\( T^{3} + 12 T^{2} + \cdots - 928 \)
|
| $73$ |
\( T^{3} - 4 T^{2} + \cdots + 326 \)
|
| $79$ |
\( T^{3} - 26 T^{2} + \cdots - 256 \)
|
| $83$ |
\( T^{3} - 6 T^{2} + \cdots + 736 \)
|
| $89$ |
\( T^{3} - 18 T^{2} + \cdots + 1024 \)
|
| $97$ |
\( T^{3} + 8 T^{2} + \cdots - 128 \)
|
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