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: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.310257.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 6x^{3} + 5x^{2} + 7x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{5} - x^{4} - 6x^{3} + 5x^{2} + 7x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} - \nu^{2} + 5\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 5\nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} - \beta _1 + 8 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{4} - 5\beta_{3} + 10\beta_{2} - 5\beta _1 + 31 ) / 3 \)
|
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 | −3.39428 | 0 | 3.49214 | 1.00000 | 0 | −3.39428 | |||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0 | 1.00000 | −1.98031 | 0 | −0.491003 | 1.00000 | 0 | −1.98031 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 0 | 1.00000 | 0.435523 | 0 | −4.62948 | 1.00000 | 0 | 0.435523 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0 | 1.00000 | 1.07247 | 0 | 0.243901 | 1.00000 | 0 | 1.07247 | ||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 0 | 1.00000 | 2.86660 | 0 | −3.61556 | 1.00000 | 0 | 2.86660 | ||||||||||||||||||||||||||||||||||
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.u | 5 | |
| 3.b | odd | 2 | 1 | 3726.2.a.r | 5 | ||
| 9.c | even | 3 | 2 | 1242.2.e.b | 10 | ||
| 9.d | odd | 6 | 2 | 414.2.e.d | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 414.2.e.d | ✓ | 10 | 9.d | odd | 6 | 2 | |
| 1242.2.e.b | 10 | 9.c | even | 3 | 2 | ||
| 3726.2.a.r | 5 | 3.b | odd | 2 | 1 | ||
| 3726.2.a.u | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3726))\):
|
\( T_{5}^{5} + T_{5}^{4} - 12T_{5}^{3} - 5T_{5}^{2} + 25T_{5} - 9 \)
|
|
\( T_{7}^{5} + 5T_{7}^{4} - 11T_{7}^{3} - 62T_{7}^{2} - 13T_{7} + 7 \)
|
|
\( T_{11}^{5} + 11T_{11}^{4} + 15T_{11}^{3} - 178T_{11}^{2} - 575T_{11} - 291 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + T^{4} - 12 T^{3} + \cdots - 9 \)
|
| $7$ |
\( T^{5} + 5 T^{4} + \cdots + 7 \)
|
| $11$ |
\( T^{5} + 11 T^{4} + \cdots - 291 \)
|
| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 1111 \)
|
| $17$ |
\( T^{5} + T^{4} + \cdots + 2037 \)
|
| $19$ |
\( T^{5} + 3 T^{4} + \cdots - 857 \)
|
| $23$ |
\( (T + 1)^{5} \)
|
| $29$ |
\( T^{5} + 8 T^{4} + \cdots + 2931 \)
|
| $31$ |
\( T^{5} + 4 T^{4} + \cdots + 1791 \)
|
| $37$ |
\( T^{5} + 14 T^{4} + \cdots + 19189 \)
|
| $41$ |
\( T^{5} + 24 T^{4} + \cdots - 6237 \)
|
| $43$ |
\( T^{5} + 27 T^{4} + \cdots - 677 \)
|
| $47$ |
\( T^{5} + 9 T^{4} + \cdots + 37125 \)
|
| $53$ |
\( T^{5} - 13 T^{4} + \cdots - 213 \)
|
| $59$ |
\( T^{5} + 9 T^{4} + \cdots - 5157 \)
|
| $61$ |
\( T^{5} + 3 T^{4} + \cdots + 1837 \)
|
| $67$ |
\( T^{5} + 5 T^{4} + \cdots + 7 \)
|
| $71$ |
\( T^{5} + 27 T^{4} + \cdots - 21357 \)
|
| $73$ |
\( T^{5} - 17 T^{4} + \cdots - 297 \)
|
| $79$ |
\( T^{5} - 11 T^{4} + \cdots - 10953 \)
|
| $83$ |
\( T^{5} + 23 T^{4} + \cdots - 10371 \)
|
| $89$ |
\( T^{5} + 39 T^{4} + \cdots - 62667 \)
|
| $97$ |
\( T^{5} + 28 T^{4} + \cdots - 4473 \)
|
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