Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3645,2,Mod(1,3645)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3645, 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("3645.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3645 = 3^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3645.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: | \(29.1054715368\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3916917.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 6x^{4} + 12x^{3} + 18x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 6x^{4} + 12x^{3} + 18x^{2} - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 3\nu^{3} - 2\nu^{2} + 6\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 4\nu^{4} - 2\nu^{3} + 13\nu^{2} + 6\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 7\beta _1 + 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 3\beta_{3} + 8\beta_{2} + 19\beta _1 + 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} + 14\beta_{3} + 23\beta_{2} + 58\beta _1 + 53 \)
|
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 |
|
−2.07036 | 0 | 2.28641 | −1.00000 | 0 | −3.94975 | −0.592975 | 0 | 2.07036 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.75528 | 0 | 1.08100 | −1.00000 | 0 | −0.223190 | 1.61309 | 0 | 1.75528 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.626526 | 0 | −1.60747 | −1.00000 | 0 | 0.973822 | −2.26017 | 0 | −0.626526 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.53828 | 0 | 0.366293 | −1.00000 | 0 | −0.341109 | −2.51309 | 0 | −1.53828 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.25286 | 0 | 3.07538 | −1.00000 | 0 | 2.60016 | 2.42267 | 0 | −2.25286 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.40798 | 0 | 3.79838 | −1.00000 | 0 | 3.94007 | 4.33047 | 0 | −2.40798 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 3645.2.a.d | yes | 6 |
| 3.b | odd | 2 | 1 | 3645.2.a.c | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3645.2.a.c | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 3645.2.a.d | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} - 6T_{2}^{4} + 22T_{2}^{3} + 3T_{2}^{2} - 39T_{2} + 19 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3645))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 19 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots - 3 \)
|
| $11$ |
\( T^{6} - 9 T^{5} + \cdots + 1 \)
|
| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 683 \)
|
| $17$ |
\( T^{6} + 15 T^{5} + \cdots + 127 \)
|
| $19$ |
\( T^{6} + 9 T^{5} + \cdots + 37 \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots - 381 \)
|
| $29$ |
\( T^{6} - 30 T^{5} + \cdots + 6823 \)
|
| $31$ |
\( T^{6} + 3 T^{5} + \cdots + 10377 \)
|
| $37$ |
\( T^{6} + 18 T^{5} + \cdots + 361 \)
|
| $41$ |
\( T^{6} - 15 T^{5} + \cdots + 6661 \)
|
| $43$ |
\( T^{6} + 3 T^{5} + \cdots + 4591 \)
|
| $47$ |
\( T^{6} - 15 T^{5} + \cdots + 10009 \)
|
| $53$ |
\( T^{6} - 33 T^{5} + \cdots - 1961 \)
|
| $59$ |
\( T^{6} - 27 T^{5} + \cdots - 31337 \)
|
| $61$ |
\( T^{6} - 3 T^{5} + \cdots - 17 \)
|
| $67$ |
\( T^{6} + 33 T^{5} + \cdots + 61881 \)
|
| $71$ |
\( T^{6} - 27 T^{5} + \cdots - 11789 \)
|
| $73$ |
\( T^{6} + 9 T^{5} + \cdots + 77689 \)
|
| $79$ |
\( T^{6} - 18 T^{5} + \cdots + 58803 \)
|
| $83$ |
\( T^{6} - 15 T^{5} + \cdots - 86661 \)
|
| $89$ |
\( T^{6} + 15 T^{5} + \cdots + 35001 \)
|
| $97$ |
\( T^{6} + 15 T^{5} + \cdots - 25973 \)
|
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