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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 75x^{2} - 42x + 736 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 69) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 75x^{2} - 42x + 736 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 54\nu + 4 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 38 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 38 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 55\beta _1 + 34 \)
|
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 |
|
−8.50608 | 0 | 40.3535 | 86.6918 | 0 | −64.0051 | −71.0553 | 0 | −737.408 | ||||||||||||||||||||||||||||||
| 1.2 | −4.86863 | 0 | −8.29644 | −66.7344 | 0 | −185.299 | 196.188 | 0 | 324.905 | |||||||||||||||||||||||||||||||
| 1.3 | 2.04157 | 0 | −27.8320 | −42.3660 | 0 | 191.647 | −122.151 | 0 | −86.4934 | |||||||||||||||||||||||||||||||
| 1.4 | 7.33314 | 0 | 21.7749 | 0.408582 | 0 | −4.34307 | −74.9818 | 0 | 2.99619 | |||||||||||||||||||||||||||||||
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.e | 4 | |
| 3.b | odd | 2 | 1 | 69.6.a.d | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1104.6.a.o | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.6.a.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 207.6.a.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 1104.6.a.o | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 4T_{2}^{3} - 69T_{2}^{2} - 188T_{2} + 620 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 4 T^{3} + \cdots + 620 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 22 T^{3} + \cdots + 100144 \)
|
| $7$ |
\( T^{4} + 62 T^{3} + \cdots - 9871616 \)
|
| $11$ |
\( T^{4} + \cdots - 45159083072 \)
|
| $13$ |
\( T^{4} + \cdots + 16813724400 \)
|
| $17$ |
\( T^{4} + \cdots - 95458629376 \)
|
| $19$ |
\( T^{4} + \cdots + 3908943190016 \)
|
| $23$ |
\( (T - 529)^{4} \)
|
| $29$ |
\( T^{4} + \cdots + 61820529282864 \)
|
| $31$ |
\( T^{4} + \cdots + 378047008189440 \)
|
| $37$ |
\( T^{4} + \cdots - 684323468629888 \)
|
| $41$ |
\( T^{4} + \cdots - 12\!\cdots\!64 \)
|
| $43$ |
\( T^{4} + \cdots + 13\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!92 \)
|
| $53$ |
\( T^{4} + \cdots + 40\!\cdots\!32 \)
|
| $59$ |
\( T^{4} + \cdots + 23\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots + 60\!\cdots\!56 \)
|
| $67$ |
\( T^{4} + \cdots - 73\!\cdots\!28 \)
|
| $71$ |
\( T^{4} + \cdots + 12\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 14\!\cdots\!88 \)
|
| $79$ |
\( T^{4} + \cdots + 75\!\cdots\!80 \)
|
| $83$ |
\( T^{4} + \cdots + 12\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots - 75\!\cdots\!28 \)
|
| $97$ |
\( T^{4} + \cdots + 20\!\cdots\!20 \)
|
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