Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [207,6,Mod(1,207)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("207.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 207 = 3^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 207.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: | \(33.1994507013\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.7925.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 13x + 12 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 23) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 13x + 12 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} - \beta _1 + 17 ) / 2 \)
|
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 |
|
−5.31473 | 0 | −3.75366 | 23.8768 | 0 | −11.7843 | 190.021 | 0 | −126.899 | |||||||||||||||||||||||||||
| 1.2 | 0.164504 | 0 | −31.9729 | 37.9865 | 0 | −43.8366 | −10.5238 | 0 | 6.24894 | ||||||||||||||||||||||||||||
| 1.3 | 9.15022 | 0 | 51.7266 | −3.86330 | 0 | −226.379 | 180.503 | 0 | −35.3501 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 207.6.a.b | 3 | |
| 3.b | odd | 2 | 1 | 23.6.a.a | ✓ | 3 | |
| 12.b | even | 2 | 1 | 368.6.a.e | 3 | ||
| 15.d | odd | 2 | 1 | 575.6.a.b | 3 | ||
| 69.c | even | 2 | 1 | 529.6.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.6.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 207.6.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
| 368.6.a.e | 3 | 12.b | even | 2 | 1 | ||
| 529.6.a.a | 3 | 69.c | even | 2 | 1 | ||
| 575.6.a.b | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4T_{2}^{2} - 48T_{2} + 8 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 4 T^{2} + \cdots + 8 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} - 58 T^{2} + \cdots + 3504 \)
|
| $7$ |
\( T^{3} + 282 T^{2} + \cdots + 116944 \)
|
| $11$ |
\( T^{3} + 136 T^{2} + \cdots - 840152 \)
|
| $13$ |
\( T^{3} + 1116 T^{2} + \cdots - 255615102 \)
|
| $17$ |
\( T^{3} - 896 T^{2} + \cdots - 220718408 \)
|
| $19$ |
\( T^{3} - 1654 T^{2} + \cdots + 460771768 \)
|
| $23$ |
\( (T - 529)^{3} \)
|
| $29$ |
\( T^{3} + \cdots + 33789223458 \)
|
| $31$ |
\( T^{3} + \cdots - 117638912880 \)
|
| $37$ |
\( T^{3} + \cdots + 1048082031344 \)
|
| $41$ |
\( T^{3} + \cdots + 1564944049486 \)
|
| $43$ |
\( T^{3} + \cdots - 95315904000 \)
|
| $47$ |
\( T^{3} + \cdots - 916008439440 \)
|
| $53$ |
\( T^{3} + \cdots - 7849670295504 \)
|
| $59$ |
\( T^{3} + \cdots + 1725012447168 \)
|
| $61$ |
\( T^{3} + \cdots + 11439907465152 \)
|
| $67$ |
\( T^{3} + \cdots - 11971711378840 \)
|
| $71$ |
\( T^{3} + \cdots + 24837760695040 \)
|
| $73$ |
\( T^{3} + \cdots + 7199078503954 \)
|
| $79$ |
\( T^{3} + \cdots + 235690469012368 \)
|
| $83$ |
\( T^{3} + \cdots + 297029282761704 \)
|
| $89$ |
\( T^{3} + \cdots - 125799322340896 \)
|
| $97$ |
\( T^{3} + \cdots + 693755159518744 \)
|
| show more | |
| show less |