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: | yes |
| Analytic conductor: | \(30.9537580313\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | 7.7.294755098673.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | no (minimal twist has level 71) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - 14x^{5} + 56x^{3} - 56x - 21 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{5} - 10\nu^{3} + \nu^{2} + 20\nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{4} + 3\nu^{3} + 8\nu^{2} - 18\nu - 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{5} + 20\nu^{3} + 6\nu^{2} - 40\nu - 24 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 12\nu^{4} + 36\nu^{2} - 5\nu - 16 \)
|
| \(\beta_{6}\) | \(=\) |
\( 4\nu^{4} - 4\nu^{3} - 32\nu^{2} + 24\nu + 32 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 2\beta_{2} + 32 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{6} + 4\beta_{3} + 6\beta_1 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{6} + 8\beta_{4} + 4\beta_{3} + 16\beta_{2} + 192 ) / 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 10\beta_{6} - \beta_{4} + 40\beta_{3} + 6\beta_{2} + 40\beta_1 ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 36\beta_{6} + 8\beta_{5} + 60\beta_{4} + 48\beta_{3} + 120\beta_{2} + 5\beta _1 + 1280 ) / 8 \)
|
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 | −5.99196 | 0 | −2.05684 | 0 | 0 | 0 | 26.9036 | 0 | |||||||||||||||||||||||||||||||||||||||
| 993.2 | 0 | −3.97864 | 0 | 6.09921 | 0 | 0 | 0 | 6.82955 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 993.3 | 0 | −3.49322 | 0 | −2.39292 | 0 | 0 | 0 | 3.20258 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 993.4 | 0 | 1.03068 | 0 | −8.93357 | 0 | 0 | 0 | −7.93769 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 993.5 | 0 | 1.63599 | 0 | 6.36873 | 0 | 0 | 0 | −6.32354 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 993.6 | 0 | 5.26388 | 0 | 9.99851 | 0 | 0 | 0 | 18.7084 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 993.7 | 0 | 5.53327 | 0 | −9.08313 | 0 | 0 | 0 | 21.6170 | 0 | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 71.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-71}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1136.3.h.b | 7 | |
| 4.b | odd | 2 | 1 | 71.3.b.b | ✓ | 7 | |
| 12.b | even | 2 | 1 | 639.3.d.b | 7 | ||
| 71.b | odd | 2 | 1 | CM | 1136.3.h.b | 7 | |
| 284.c | even | 2 | 1 | 71.3.b.b | ✓ | 7 | |
| 852.d | odd | 2 | 1 | 639.3.d.b | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 71.3.b.b | ✓ | 7 | 4.b | odd | 2 | 1 | |
| 71.3.b.b | ✓ | 7 | 284.c | even | 2 | 1 | |
| 639.3.d.b | 7 | 12.b | even | 2 | 1 | ||
| 639.3.d.b | 7 | 852.d | odd | 2 | 1 | ||
| 1136.3.h.b | 7 | 1.a | even | 1 | 1 | trivial | |
| 1136.3.h.b | 7 | 71.b | odd | 2 | 1 | CM | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 63T_{3}^{5} + 1134T_{3}^{3} - 5103T_{3} + 4090 \)
acting on \(S_{3}^{\mathrm{new}}(1136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} \)
|
| $3$ |
\( T^{7} - 63 T^{5} + \cdots + 4090 \)
|
| $5$ |
\( T^{7} - 175 T^{5} + \cdots - 155114 \)
|
| $7$ |
\( T^{7} \)
|
| $11$ |
\( T^{7} \)
|
| $13$ |
\( T^{7} \)
|
| $17$ |
\( T^{7} \)
|
| $19$ |
\( T^{7} + \cdots - 1681323622 \)
|
| $23$ |
\( T^{7} \)
|
| $29$ |
\( T^{7} + \cdots + 33116358982 \)
|
| $31$ |
\( T^{7} \)
|
| $37$ |
\( T^{7} + \cdots + 14146851670 \)
|
| $41$ |
\( T^{7} \)
|
| $43$ |
\( T^{7} + \cdots - 305153453110 \)
|
| $47$ |
\( T^{7} \)
|
| $53$ |
\( T^{7} \)
|
| $59$ |
\( T^{7} \)
|
| $61$ |
\( T^{7} \)
|
| $67$ |
\( T^{7} \)
|
| $71$ |
\( (T - 71)^{7} \)
|
| $73$ |
\( T^{7} + \cdots + 16831893051790 \)
|
| $79$ |
\( T^{7} + \cdots + 38386837749218 \)
|
| $83$ |
\( T^{7} + \cdots + 21130236657050 \)
|
| $89$ |
\( T^{7} + \cdots + 8253987626158 \)
|
| $97$ |
\( T^{7} \)
|
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