Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2009,2,Mod(1,2009)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, 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("2009.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2009 = 7^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2009.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: | \(16.0419457661\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.633117.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 4x^{2} + 6x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 287) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 4x^{2} + 6x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{4} + \nu^{3} + 5\nu^{2} - 3\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta_{2} + \beta _1 + 14 \)
|
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 |
|
−2.45719 | 1.45719 | 4.03778 | 2.26685 | −3.58059 | 0 | −5.00722 | −0.876597 | −5.57007 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.20098 | 0.200978 | −0.557652 | 3.21704 | −0.241370 | 0 | 3.07168 | −2.95961 | −3.86360 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.460315 | −0.539685 | −1.78811 | −4.10136 | 0.248425 | 0 | 1.74372 | −2.70874 | 1.88791 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.08727 | −2.08727 | −0.817843 | −0.209668 | −2.26943 | 0 | −3.06376 | 1.35670 | −0.227965 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.03121 | −3.03121 | 2.12582 | 3.82713 | −6.15703 | 0 | 0.255573 | 6.18825 | 7.77372 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(41\) | \(-1\) |
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 | 2009.2.a.n | 5 | |
| 7.b | odd | 2 | 1 | 287.2.a.e | ✓ | 5 | |
| 21.c | even | 2 | 1 | 2583.2.a.r | 5 | ||
| 28.d | even | 2 | 1 | 4592.2.a.bb | 5 | ||
| 35.c | odd | 2 | 1 | 7175.2.a.n | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 287.2.a.e | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 2009.2.a.n | 5 | 1.a | even | 1 | 1 | trivial | |
| 2583.2.a.r | 5 | 21.c | even | 2 | 1 | ||
| 4592.2.a.bb | 5 | 28.d | even | 2 | 1 | ||
| 7175.2.a.n | 5 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\):
|
\( T_{2}^{5} + T_{2}^{4} - 6T_{2}^{3} - 4T_{2}^{2} + 6T_{2} + 3 \)
|
|
\( T_{3}^{5} + 4T_{3}^{4} - 10T_{3}^{2} - 3T_{3} + 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 6 T^{3} + \cdots + 3 \)
|
| $3$ |
\( T^{5} + 4 T^{4} + \cdots + 1 \)
|
| $5$ |
\( T^{5} - 5 T^{4} + \cdots - 24 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 2 T^{4} + \cdots - 2472 \)
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| $13$ |
\( T^{5} + 5 T^{4} + \cdots - 49 \)
|
| $17$ |
\( T^{5} + 13 T^{4} + \cdots - 2049 \)
|
| $19$ |
\( T^{5} - 48 T^{3} + \cdots + 1 \)
|
| $23$ |
\( T^{5} - 2 T^{4} + \cdots + 1317 \)
|
| $29$ |
\( T^{5} + 5 T^{4} + \cdots + 1512 \)
|
| $31$ |
\( T^{5} + 17 T^{4} + \cdots + 56 \)
|
| $37$ |
\( T^{5} + 7 T^{4} + \cdots + 4 \)
|
| $41$ |
\( (T - 1)^{5} \)
|
| $43$ |
\( T^{5} - T^{4} + \cdots - 1751 \)
|
| $47$ |
\( T^{5} + 9 T^{4} + \cdots + 10092 \)
|
| $53$ |
\( T^{5} - 5 T^{4} + \cdots + 2328 \)
|
| $59$ |
\( T^{5} + 7 T^{4} + \cdots + 21000 \)
|
| $61$ |
\( T^{5} + 22 T^{4} + \cdots - 2504 \)
|
| $67$ |
\( T^{5} + 3 T^{4} + \cdots - 472 \)
|
| $71$ |
\( T^{5} + 24 T^{4} + \cdots + 43128 \)
|
| $73$ |
\( T^{5} + 40 T^{4} + \cdots + 12184 \)
|
| $79$ |
\( T^{5} + 42 T^{4} + \cdots - 75008 \)
|
| $83$ |
\( T^{5} - 12 T^{4} + \cdots - 24696 \)
|
| $89$ |
\( T^{5} + 8 T^{4} + \cdots - 3 \)
|
| $97$ |
\( T^{5} + 16 T^{4} + \cdots + 10493 \)
|
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