Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,3,Mod(91,207)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("207.91");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.d (of order \(2\), degree \(1\), 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: | \(5.64034147226\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.621.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 23) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 3\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(47\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 91.1 |
|
−3.12398 | 0 | 5.75927 | 0 | 0 | 0 | −5.49593 | 0 | 0 | |||||||||||||||||||||||||||
| 91.2 | −0.601466 | 0 | −3.63824 | 0 | 0 | 0 | 4.59414 | 0 | 0 | ||||||||||||||||||||||||||||
| 91.3 | 3.72545 | 0 | 9.87897 | 0 | 0 | 0 | 21.9018 | 0 | 0 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 207.3.d.a | 3 | |
| 3.b | odd | 2 | 1 | 23.3.b.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 368.3.f.a | 3 | ||
| 15.d | odd | 2 | 1 | 575.3.d.b | 3 | ||
| 15.e | even | 4 | 2 | 575.3.c.a | 6 | ||
| 23.b | odd | 2 | 1 | CM | 207.3.d.a | 3 | |
| 24.f | even | 2 | 1 | 1472.3.f.b | 3 | ||
| 24.h | odd | 2 | 1 | 1472.3.f.a | 3 | ||
| 69.c | even | 2 | 1 | 23.3.b.a | ✓ | 3 | |
| 276.h | odd | 2 | 1 | 368.3.f.a | 3 | ||
| 345.h | even | 2 | 1 | 575.3.d.b | 3 | ||
| 345.l | odd | 4 | 2 | 575.3.c.a | 6 | ||
| 552.b | even | 2 | 1 | 1472.3.f.a | 3 | ||
| 552.h | odd | 2 | 1 | 1472.3.f.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.3.b.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 23.3.b.a | ✓ | 3 | 69.c | even | 2 | 1 | |
| 207.3.d.a | 3 | 1.a | even | 1 | 1 | trivial | |
| 207.3.d.a | 3 | 23.b | odd | 2 | 1 | CM | |
| 368.3.f.a | 3 | 12.b | even | 2 | 1 | ||
| 368.3.f.a | 3 | 276.h | odd | 2 | 1 | ||
| 575.3.c.a | 6 | 15.e | even | 4 | 2 | ||
| 575.3.c.a | 6 | 345.l | odd | 4 | 2 | ||
| 575.3.d.b | 3 | 15.d | odd | 2 | 1 | ||
| 575.3.d.b | 3 | 345.h | even | 2 | 1 | ||
| 1472.3.f.a | 3 | 24.h | odd | 2 | 1 | ||
| 1472.3.f.a | 3 | 552.b | even | 2 | 1 | ||
| 1472.3.f.b | 3 | 24.f | even | 2 | 1 | ||
| 1472.3.f.b | 3 | 552.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 12T_{2} - 7 \)
acting on \(S_{3}^{\mathrm{new}}(207, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 12T - 7 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} - 507T - 1082 \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} \)
|
| $23$ |
\( (T - 23)^{3} \)
|
| $29$ |
\( T^{3} - 2523T + 30746 \)
|
| $31$ |
\( T^{3} - 2883T - 58754 \)
|
| $37$ |
\( T^{3} \)
|
| $41$ |
\( T^{3} - 5043T + 43634 \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( T^{3} - 6627 T - 205342 \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( (T + 26)^{3} \)
|
| $61$ |
\( T^{3} \)
|
| $67$ |
\( T^{3} \)
|
| $71$ |
\( T^{3} - 15123 T + 667154 \)
|
| $73$ |
\( T^{3} - 15987 T - 725042 \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( T^{3} \)
|
| $97$ |
\( T^{3} \)
|
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