Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1144,2,Mod(1,1144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1144, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1144.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1144 = 2^{3} \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1144.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: | \(9.13488599123\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 14x^{4} + 11x^{3} + 41x^{2} - 38x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 14x^{4} + 11x^{3} + 41x^{2} - 38x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 14\nu^{3} + 9\nu^{2} + 43\nu - 22 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 16\nu^{3} - \nu^{2} - 54\nu + 22 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} + \nu^{4} + 28\nu^{3} - 6\nu^{2} - 81\nu + 26 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{5} - \nu^{4} - 28\nu^{3} + 8\nu^{2} + 81\nu - 36 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 12\beta_{5} + 10\beta_{4} - 4\beta_{2} + 5\beta _1 + 42 \)
|
| \(\nu^{5}\) | \(=\) |
\( 31\beta_{5} + 15\beta_{4} + 14\beta_{3} - 16\beta_{2} + 74\beta _1 + 33 \)
|
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 | −3.39006 | 0 | 2.28239 | 0 | 1.53655 | 0 | 8.49253 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.88556 | 0 | −3.20883 | 0 | −3.75272 | 0 | 0.555345 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.546138 | 0 | 0.0762851 | 0 | 3.03523 | 0 | −2.70173 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.347518 | 0 | 4.28345 | 0 | −4.03224 | 0 | −2.87923 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.29108 | 0 | −1.20574 | 0 | 1.67022 | 0 | 2.24907 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.87820 | 0 | 2.77246 | 0 | 0.542969 | 0 | 5.28402 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 1144.2.a.l | ✓ | 6 |
| 4.b | odd | 2 | 1 | 2288.2.a.ba | 6 | ||
| 8.b | even | 2 | 1 | 9152.2.a.cq | 6 | ||
| 8.d | odd | 2 | 1 | 9152.2.a.co | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1144.2.a.l | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 2288.2.a.ba | 6 | 4.b | odd | 2 | 1 | ||
| 9152.2.a.co | 6 | 8.d | odd | 2 | 1 | ||
| 9152.2.a.cq | 6 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1144))\):
|
\( T_{3}^{6} + T_{3}^{5} - 14T_{3}^{4} - 11T_{3}^{3} + 41T_{3}^{2} + 38T_{3} + 8 \)
|
|
\( T_{5}^{6} - 5T_{5}^{5} - 9T_{5}^{4} + 61T_{5}^{3} - 16T_{5}^{2} - 104T_{5} + 8 \)
|
|
\( T_{7}^{6} + T_{7}^{5} - 22T_{7}^{4} + 5T_{7}^{3} + 129T_{7}^{2} - 186T_{7} + 64 \)
|
|
\( T_{17}^{6} + T_{17}^{5} - 60T_{17}^{4} - 80T_{17}^{3} + 672T_{17}^{2} + 1200T_{17} + 512 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} - 14 T^{4} + \cdots + 8 \)
|
| $5$ |
\( T^{6} - 5 T^{5} + \cdots + 8 \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots + 64 \)
|
| $11$ |
\( (T - 1)^{6} \)
|
| $13$ |
\( (T - 1)^{6} \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots + 512 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots - 512 \)
|
| $23$ |
\( T^{6} + 3 T^{5} + \cdots + 1088 \)
|
| $29$ |
\( T^{6} - 17 T^{5} + \cdots - 2816 \)
|
| $31$ |
\( T^{6} + T^{5} + \cdots - 2048 \)
|
| $37$ |
\( T^{6} - 26 T^{5} + \cdots - 71872 \)
|
| $41$ |
\( T^{6} - 16 T^{5} + \cdots + 2368 \)
|
| $43$ |
\( T^{6} + 18 T^{5} + \cdots + 18496 \)
|
| $47$ |
\( T^{6} - 11 T^{5} + \cdots - 5632 \)
|
| $53$ |
\( T^{6} - 27 T^{5} + \cdots - 13456 \)
|
| $59$ |
\( T^{6} - 16 T^{5} + \cdots + 404896 \)
|
| $61$ |
\( T^{6} - 17 T^{5} + \cdots - 2816 \)
|
| $67$ |
\( T^{6} - 8 T^{5} + \cdots - 4352 \)
|
| $71$ |
\( T^{6} - 2 T^{5} + \cdots + 4096 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots - 675968 \)
|
| $79$ |
\( T^{6} + 2 T^{5} + \cdots - 262144 \)
|
| $83$ |
\( T^{6} + 17 T^{5} + \cdots + 334336 \)
|
| $89$ |
\( T^{6} - 5 T^{5} + \cdots - 47872 \)
|
| $97$ |
\( T^{6} - 9 T^{5} + \cdots - 512 \)
|
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