Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1136,3,Mod(993,1136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1136, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1136.993");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1136 = 2^{4} \cdot 71 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1136.h (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(30.9537580313\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.2836736.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 108x^{2} - 40x + 2825 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 71) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 108x^{2} - 40x + 2825 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 10\nu^{2} - 53\nu + 570 ) / 155 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 10\nu^{2} + 363\nu - 570 ) / 155 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 11\nu^{2} + 106\nu + 534 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 10\beta _1 - 54 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 20\beta_{3} - 53\beta_{2} - 163\beta _1 + 60 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1136\mathbb{Z}\right)^\times\).
| \(n\) | \(143\) | \(433\) | \(853\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 993.1 |
|
0 | −0.414214 | 0 | −0.585786 | 0 | − | 8.98419i | 0 | −8.82843 | 0 | |||||||||||||||||||||||||||||
| 993.2 | 0 | −0.414214 | 0 | −0.585786 | 0 | 8.98419i | 0 | −8.82843 | 0 | |||||||||||||||||||||||||||||||
| 993.3 | 0 | 2.41421 | 0 | −3.41421 | 0 | − | 11.7168i | 0 | −3.17157 | 0 | ||||||||||||||||||||||||||||||
| 993.4 | 0 | 2.41421 | 0 | −3.41421 | 0 | 11.7168i | 0 | −3.17157 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 71.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1136.3.h.a | 4 | |
| 4.b | odd | 2 | 1 | 71.3.b.a | ✓ | 4 | |
| 12.b | even | 2 | 1 | 639.3.d.a | 4 | ||
| 71.b | odd | 2 | 1 | inner | 1136.3.h.a | 4 | |
| 284.c | even | 2 | 1 | 71.3.b.a | ✓ | 4 | |
| 852.d | odd | 2 | 1 | 639.3.d.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.3.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 71.3.b.a | ✓ | 4 | 284.c | even | 2 | 1 | |
| 639.3.d.a | 4 | 12.b | even | 2 | 1 | ||
| 639.3.d.a | 4 | 852.d | odd | 2 | 1 | ||
| 1136.3.h.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 1136.3.h.a | 4 | 71.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 2T_{3} - 1 \)
acting on \(S_{3}^{\mathrm{new}}(1136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 2 T - 1)^{2} \)
|
| $5$ |
\( (T^{2} + 4 T + 2)^{2} \)
|
| $7$ |
\( T^{4} + 218 T^{2} + 11081 \)
|
| $11$ |
\( T^{4} + 436 T^{2} + 44324 \)
|
| $13$ |
\( T^{4} + 494 T^{2} + 11081 \)
|
| $17$ |
\( T^{4} + 436 T^{2} + 44324 \)
|
| $19$ |
\( (T^{2} + 6 T - 233)^{2} \)
|
| $23$ |
\( T^{4} + 1642 T^{2} + 542969 \)
|
| $29$ |
\( (T^{2} - 98)^{2} \)
|
| $31$ |
\( T^{4} + 1962 T^{2} + 897561 \)
|
| $37$ |
\( (T^{2} + 12 T - 2702)^{2} \)
|
| $41$ |
\( T^{4} + 3836 T^{2} + 2171876 \)
|
| $43$ |
\( (T^{2} - 30 T - 497)^{2} \)
|
| $47$ |
\( T^{4} + 5450 T^{2} + 6925625 \)
|
| $53$ |
\( T^{4} + 6512 T^{2} + 709184 \)
|
| $59$ |
\( T^{4} + 3488 T^{2} + 2836736 \)
|
| $61$ |
\( T^{4} + 11512 T^{2} + 8687504 \)
|
| $67$ |
\( T^{4} + 7324 T^{2} + 12809636 \)
|
| $71$ |
\( T^{4} + 152 T^{3} + \cdots + 25411681 \)
|
| $73$ |
\( (T^{2} - 34 T - 6439)^{2} \)
|
| $79$ |
\( (T^{2} + 96 T + 126)^{2} \)
|
| $83$ |
\( (T^{2} + 164 T + 5572)^{2} \)
|
| $89$ |
\( (T^{2} + 94 T + 1241)^{2} \)
|
| $97$ |
\( T^{4} + 21364 T^{2} + 106421924 \)
|
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