Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1665,2,Mod(1,1665)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1665, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("1665.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1665 = 3^{2} \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1665.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: | \(13.2950919365\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.95034688.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 16x^{2} - 12x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - 2x^{5} - 8x^{4} + 12x^{3} + 16x^{2} - 12x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - \nu^{4} - 9\nu^{3} + 5\nu^{2} + 19\nu - 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 11\nu^{2} + 17\nu - 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 6\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + 10\beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{5} + 2\beta_{4} + 10\beta_{3} + 11\beta_{2} + 40\beta _1 + 12 \)
|
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 |
|
−2.76746 | 0 | 5.65882 | −1.00000 | 0 | 0.699359 | −10.1256 | 0 | 2.76746 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.28339 | 0 | 3.21388 | −1.00000 | 0 | −3.72551 | −2.77177 | 0 | 2.28339 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.505405 | 0 | −1.74457 | −1.00000 | 0 | −1.65336 | 1.89252 | 0 | 0.505405 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.0966244 | 0 | −1.99066 | −1.00000 | 0 | 0.508444 | 0.385595 | 0 | 0.0966244 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.51760 | 0 | 0.303121 | −1.00000 | 0 | 2.78967 | −2.57519 | 0 | −1.51760 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.13528 | 0 | 2.55940 | −1.00000 | 0 | −2.61861 | 1.19448 | 0 | −2.13528 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(37\) | \( +1 \) |
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 | 1665.2.a.t | ✓ | 6 |
| 3.b | odd | 2 | 1 | 1665.2.a.u | yes | 6 | |
| 5.b | even | 2 | 1 | 8325.2.a.ck | 6 | ||
| 15.d | odd | 2 | 1 | 8325.2.a.cj | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1665.2.a.t | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1665.2.a.u | yes | 6 | 3.b | odd | 2 | 1 | |
| 8325.2.a.cj | 6 | 15.d | odd | 2 | 1 | ||
| 8325.2.a.ck | 6 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1665))\):
|
\( T_{2}^{6} + 2T_{2}^{5} - 8T_{2}^{4} - 12T_{2}^{3} + 16T_{2}^{2} + 12T_{2} + 1 \)
|
|
\( T_{7}^{6} + 4T_{7}^{5} - 8T_{7}^{4} - 36T_{7}^{3} + 3T_{7}^{2} + 40T_{7} - 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} + \cdots + 1 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( T^{6} + 4 T^{5} + \cdots - 16 \)
|
| $11$ |
\( T^{6} + 4 T^{5} + \cdots + 1008 \)
|
| $13$ |
\( T^{6} + 6 T^{5} + \cdots - 644 \)
|
| $17$ |
\( T^{6} + 8 T^{5} + \cdots - 28 \)
|
| $19$ |
\( T^{6} - 16 T^{5} + \cdots - 716 \)
|
| $23$ |
\( T^{6} + 10 T^{5} + \cdots - 32 \)
|
| $29$ |
\( T^{6} + 6 T^{5} + \cdots + 1036 \)
|
| $31$ |
\( T^{6} - 14 T^{5} + \cdots + 84 \)
|
| $37$ |
\( (T + 1)^{6} \)
|
| $41$ |
\( T^{6} + 20 T^{5} + \cdots + 128 \)
|
| $43$ |
\( T^{6} + 14 T^{5} + \cdots - 86212 \)
|
| $47$ |
\( T^{6} + 8 T^{5} + \cdots + 592 \)
|
| $53$ |
\( T^{6} + 12 T^{5} + \cdots + 435456 \)
|
| $59$ |
\( T^{6} + 12 T^{5} + \cdots - 336 \)
|
| $61$ |
\( T^{6} + 10 T^{5} + \cdots + 1876 \)
|
| $67$ |
\( T^{6} + 4 T^{5} + \cdots - 21904 \)
|
| $71$ |
\( T^{6} + 16 T^{5} + \cdots + 5808 \)
|
| $73$ |
\( T^{6} + 12 T^{5} + \cdots + 6476 \)
|
| $79$ |
\( T^{6} + 6 T^{5} + \cdots - 5628 \)
|
| $83$ |
\( T^{6} - 6 T^{5} + \cdots - 61056 \)
|
| $89$ |
\( T^{6} + 6 T^{5} + \cdots + 12748 \)
|
| $97$ |
\( T^{6} + 22 T^{5} + \cdots - 61348 \)
|
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