Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1127,2,Mod(1,1127)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1127, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1127.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1127 = 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1127.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: | \(8.99914030780\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.89672832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 9x^{4} + 21x^{2} - 8 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{6} - 9x^{4} + 21x^{2} - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 9\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{4} + 7\beta_{3} + 19\beta_1 \)
|
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 |
|
−2.36147 | −0.248991 | 3.57653 | 3.40145 | 0.587984 | 0 | −3.72294 | −2.93800 | −8.03242 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.36147 | 0.248991 | 3.57653 | −3.40145 | −0.587984 | 0 | −3.72294 | −2.93800 | 8.03242 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.167449 | −2.79366 | −1.97196 | 4.27537 | 0.467795 | 0 | 0.665102 | 4.80451 | −0.715908 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.167449 | 2.79366 | −1.97196 | −4.27537 | −0.467795 | 0 | 0.665102 | 4.80451 | 0.715908 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.52892 | −2.03310 | 4.39543 | −0.388988 | −5.14154 | 0 | 6.05784 | 1.13349 | −0.983720 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.52892 | 2.03310 | 4.39543 | 0.388988 | 5.14154 | 0 | 6.05784 | 1.13349 | 0.983720 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1127.2.a.j | ✓ | 6 |
| 7.b | odd | 2 | 1 | inner | 1127.2.a.j | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1127.2.a.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1127.2.a.j | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1127))\):
|
\( T_{2}^{3} - 6T_{2} - 1 \)
|
|
\( T_{3}^{6} - 12T_{3}^{4} + 33T_{3}^{2} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{3} - 6 T - 1)^{2} \)
|
| $3$ |
\( T^{6} - 12 T^{4} + \cdots - 2 \)
|
| $5$ |
\( T^{6} - 30 T^{4} + \cdots - 32 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T - 2)^{6} \)
|
| $13$ |
\( T^{6} - 66 T^{4} + \cdots - 5000 \)
|
| $17$ |
\( T^{6} - 90 T^{4} + \cdots - 12800 \)
|
| $19$ |
\( T^{6} - 84 T^{4} + \cdots - 4608 \)
|
| $23$ |
\( (T + 1)^{6} \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 3 T + 24)^{2} \)
|
| $31$ |
\( T^{6} - 132 T^{4} + \cdots - 38642 \)
|
| $37$ |
\( (T^{3} - 12 T^{2} + 248)^{2} \)
|
| $41$ |
\( T^{6} - 18 T^{4} + \cdots - 128 \)
|
| $43$ |
\( (T - 4)^{6} \)
|
| $47$ |
\( T^{6} - 132 T^{4} + \cdots - 38642 \)
|
| $53$ |
\( (T^{3} - 12 T^{2} + \cdots + 904)^{2} \)
|
| $59$ |
\( T^{6} - 306 T^{4} + \cdots - 301088 \)
|
| $61$ |
\( T^{6} - 66 T^{4} + \cdots - 288 \)
|
| $67$ |
\( (T^{3} - 168 T + 808)^{2} \)
|
| $71$ |
\( (T^{3} - 21 T + 16)^{2} \)
|
| $73$ |
\( T^{6} - 210 T^{4} + \cdots - 102152 \)
|
| $79$ |
\( (T^{3} - 24 T^{2} + 144 T - 8)^{2} \)
|
| $83$ |
\( T^{6} - 168 T^{4} + \cdots - 32768 \)
|
| $89$ |
\( T^{6} - 426 T^{4} + \cdots - 1905152 \)
|
| $97$ |
\( T^{6} - 558 T^{4} + \cdots - 3634208 \)
|
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