Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3726,2,Mod(1,3726)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3726, 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("3726.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3726 = 2 \cdot 3^{4} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3726.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: | \(29.7522597931\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.702657.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10x^{3} - 3x^{2} + 10x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 414) |
| 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
\( x^{5} - 10x^{3} - 3x^{2} + 10x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{4} - \nu^{3} + 10\nu^{2} + 12\nu - 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{4} + \nu^{3} - 20\nu^{2} - 16\nu + 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( 4\nu^{4} + 2\nu^{3} - 39\nu^{2} - 31\nu + 24 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 2\beta_{3} - \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} - 2\beta_{2} + 8\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{4} - 19\beta_{3} + \beta_{2} - 6\beta _1 + 32 \)
|
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 |
|
−1.00000 | 0 | 1.00000 | −2.16943 | 0 | −2.82239 | −1.00000 | 0 | 2.16943 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | 0 | 1.00000 | 0.462315 | 0 | −0.330894 | −1.00000 | 0 | −0.462315 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | 0 | 1.00000 | 0.549792 | 0 | −1.41606 | −1.00000 | 0 | −0.549792 | ||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0 | 1.00000 | 2.43785 | 0 | 3.58116 | −1.00000 | 0 | −2.43785 | ||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0 | 1.00000 | 3.71947 | 0 | −4.01181 | −1.00000 | 0 | −3.71947 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(23\) | \(1\) |
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 | 3726.2.a.s | 5 | |
| 3.b | odd | 2 | 1 | 3726.2.a.t | 5 | ||
| 9.c | even | 3 | 2 | 1242.2.e.c | 10 | ||
| 9.d | odd | 6 | 2 | 414.2.e.c | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 414.2.e.c | ✓ | 10 | 9.d | odd | 6 | 2 | |
| 1242.2.e.c | 10 | 9.c | even | 3 | 2 | ||
| 3726.2.a.s | 5 | 1.a | even | 1 | 1 | trivial | |
| 3726.2.a.t | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3726))\):
|
\( T_{5}^{5} - 5T_{5}^{4} + 23T_{5}^{2} - 21T_{5} + 5 \)
|
|
\( T_{7}^{5} + 5T_{7}^{4} - 7T_{7}^{3} - 62T_{7}^{2} - 77T_{7} - 19 \)
|
|
\( T_{11}^{5} - 3T_{11}^{4} - 37T_{11}^{3} + 120T_{11}^{2} + 259T_{11} - 843 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} - 5 T^{4} + \cdots + 5 \)
|
| $7$ |
\( T^{5} + 5 T^{4} + \cdots - 19 \)
|
| $11$ |
\( T^{5} - 3 T^{4} + \cdots - 843 \)
|
| $13$ |
\( T^{5} + 8 T^{4} + \cdots - 19 \)
|
| $17$ |
\( T^{5} - T^{4} + \cdots + 501 \)
|
| $19$ |
\( T^{5} + T^{4} + \cdots + 2043 \)
|
| $23$ |
\( (T + 1)^{5} \)
|
| $29$ |
\( T^{5} - 18 T^{4} + \cdots + 681 \)
|
| $31$ |
\( T^{5} + 8 T^{4} + \cdots + 10609 \)
|
| $37$ |
\( T^{5} + 6 T^{4} + \cdots + 2043 \)
|
| $41$ |
\( T^{5} - 24 T^{4} + \cdots + 12551 \)
|
| $43$ |
\( T^{5} - 11 T^{4} + \cdots - 1029 \)
|
| $47$ |
\( T^{5} - 9 T^{4} + \cdots - 553 \)
|
| $53$ |
\( T^{5} - 29 T^{4} + \cdots - 3947 \)
|
| $59$ |
\( T^{5} - 21 T^{4} + \cdots - 91617 \)
|
| $61$ |
\( T^{5} + 17 T^{4} + \cdots + 12311 \)
|
| $67$ |
\( T^{5} + 3 T^{4} + \cdots - 19845 \)
|
| $71$ |
\( T^{5} - 9 T^{4} + \cdots - 17079 \)
|
| $73$ |
\( T^{5} + 7 T^{4} + \cdots - 89 \)
|
| $79$ |
\( T^{5} + 15 T^{4} + \cdots + 33 \)
|
| $83$ |
\( T^{5} - 21 T^{4} + \cdots + 1501 \)
|
| $89$ |
\( T^{5} - 9 T^{4} + \cdots + 849 \)
|
| $97$ |
\( T^{5} - 32 T^{4} + \cdots + 33075 \)
|
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