Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,2,Mod(1,209)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("209.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 209.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: | \(1.66887340224\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.246832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 7\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 10\beta _1 + 6 \)
|
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 |
|
−2.18524 | 2.15766 | 2.77529 | −3.43077 | −4.71500 | 3.93972 | −1.69419 | 1.65548 | 7.49706 | |||||||||||||||||||||||||||||||||
| 1.2 | −0.779856 | −2.98063 | −1.39182 | −3.49235 | 2.32446 | 1.06736 | 2.64513 | 5.88418 | 2.72353 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.484093 | 2.26452 | −1.76565 | 0.637602 | 1.09624 | 2.66942 | −1.82293 | 2.12805 | 0.308658 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.82669 | −0.563416 | 1.33679 | 2.34577 | −1.02918 | 1.69239 | −1.21147 | −2.68256 | 4.28499 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.65432 | 0.121872 | 5.04540 | −1.06025 | 0.323487 | −3.36889 | 8.08346 | −2.98515 | −2.81425 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \( -1 \) |
| \(19\) | \( +1 \) |
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 | 209.2.a.c | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1881.2.a.k | 5 | ||
| 4.b | odd | 2 | 1 | 3344.2.a.t | 5 | ||
| 5.b | even | 2 | 1 | 5225.2.a.h | 5 | ||
| 11.b | odd | 2 | 1 | 2299.2.a.n | 5 | ||
| 19.b | odd | 2 | 1 | 3971.2.a.h | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.a.c | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1881.2.a.k | 5 | 3.b | odd | 2 | 1 | ||
| 2299.2.a.n | 5 | 11.b | odd | 2 | 1 | ||
| 3344.2.a.t | 5 | 4.b | odd | 2 | 1 | ||
| 3971.2.a.h | 5 | 19.b | odd | 2 | 1 | ||
| 5225.2.a.h | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 6T_{2}^{3} + 10T_{2}^{2} + 5T_{2} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(209))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 4 \)
|
| $3$ |
\( T^{5} - T^{4} - 9 T^{3} + \cdots - 1 \)
|
| $5$ |
\( T^{5} + 5 T^{4} + \cdots + 19 \)
|
| $7$ |
\( T^{5} - 6 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T - 1)^{5} \)
|
| $13$ |
\( T^{5} - 4 T^{4} + \cdots + 2 \)
|
| $17$ |
\( T^{5} + 4 T^{4} + \cdots - 64 \)
|
| $19$ |
\( (T + 1)^{5} \)
|
| $23$ |
\( T^{5} - 3 T^{4} + \cdots - 784 \)
|
| $29$ |
\( T^{5} - 10 T^{4} + \cdots + 490 \)
|
| $31$ |
\( T^{5} - 11 T^{4} + \cdots - 757 \)
|
| $37$ |
\( T^{5} - T^{4} + \cdots - 3088 \)
|
| $41$ |
\( T^{5} - 2 T^{4} + \cdots - 4112 \)
|
| $43$ |
\( T^{5} - 20 T^{4} + \cdots + 11266 \)
|
| $47$ |
\( T^{5} + 20 T^{4} + \cdots + 13184 \)
|
| $53$ |
\( T^{5} + 14 T^{4} + \cdots + 30304 \)
|
| $59$ |
\( T^{5} - 3 T^{4} + \cdots - 2000 \)
|
| $61$ |
\( T^{5} + 10 T^{4} + \cdots - 736 \)
|
| $67$ |
\( T^{5} - 9 T^{4} + \cdots + 17689 \)
|
| $71$ |
\( T^{5} - 23 T^{4} + \cdots + 19081 \)
|
| $73$ |
\( T^{5} - 340 T^{3} + \cdots + 155392 \)
|
| $79$ |
\( T^{5} - 44 T^{4} + \cdots - 36800 \)
|
| $83$ |
\( T^{5} + 14 T^{4} + \cdots - 3908 \)
|
| $89$ |
\( T^{5} + 27 T^{4} + \cdots + 320 \)
|
| $97$ |
\( T^{5} - 15 T^{4} + \cdots - 37456 \)
|
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