Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9660,2,Mod(1,9660)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9660, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("9660.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9660 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9660.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: | \(77.1354883526\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.914299812.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 14x^{4} + 42x^{2} + 12x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| 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
\( x^{6} - x^{5} - 14x^{4} + 42x^{2} + 12x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 8\nu^{3} + 16\nu^{2} + 10\nu - 12 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 8\nu^{3} + 20\nu^{2} + 6\nu - 32 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 2\nu^{4} + 11\nu^{3} - 10\nu^{2} - 20\nu + 10 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 11\nu^{3} + 10\nu^{2} + 24\nu - 10 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} + 21\nu^{3} - 32\nu^{2} - 42\nu + 38 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 + 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{5} + 9\beta_{4} + 7\beta_{3} + 6\beta_{2} - 2\beta _1 + 14 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 6\beta_{5} + 23\beta_{4} + 13\beta_{3} + 30\beta_{2} - 26\beta _1 + 98 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 34\beta_{5} + 115\beta_{4} + 69\beta_{3} + 106\beta_{2} - 54\beta _1 + 270 ) / 2 \)
|
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 |
|
0 | 1.00000 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.00000 | 0 | −1.00000 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(1\) |
| \(23\) | \(-1\) |
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 | 9660.2.a.v | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 9660.2.a.v | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{11}^{6} + 3T_{11}^{5} - 39T_{11}^{4} - 110T_{11}^{3} + 276T_{11}^{2} + 642T_{11} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9660))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( (T + 1)^{6} \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 64 \)
|
| $13$ |
\( T^{6} - 7 T^{5} + \cdots + 468 \)
|
| $17$ |
\( T^{6} + T^{5} + \cdots + 1872 \)
|
| $19$ |
\( T^{6} + 5 T^{5} + \cdots - 5832 \)
|
| $23$ |
\( (T - 1)^{6} \)
|
| $29$ |
\( T^{6} - 4 T^{5} + \cdots - 96 \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots - 96 \)
|
| $37$ |
\( T^{6} - 3 T^{5} + \cdots - 72 \)
|
| $41$ |
\( T^{6} + 3 T^{5} + \cdots + 11664 \)
|
| $43$ |
\( T^{6} + 5 T^{5} + \cdots - 1248 \)
|
| $47$ |
\( T^{6} - 14 T^{5} + \cdots - 384 \)
|
| $53$ |
\( T^{6} + 10 T^{5} + \cdots + 2496 \)
|
| $59$ |
\( T^{6} + T^{5} + \cdots - 15468 \)
|
| $61$ |
\( T^{6} - 9 T^{5} + \cdots - 81176 \)
|
| $67$ |
\( T^{6} - 13 T^{5} + \cdots - 1536 \)
|
| $71$ |
\( T^{6} - 6 T^{5} + \cdots - 118656 \)
|
| $73$ |
\( T^{6} + 3 T^{5} + \cdots + 8884 \)
|
| $79$ |
\( T^{6} - 16 T^{5} + \cdots + 1880944 \)
|
| $83$ |
\( T^{6} - 7 T^{5} + \cdots + 26064 \)
|
| $89$ |
\( T^{6} + 4 T^{5} + \cdots + 16288 \)
|
| $97$ |
\( T^{6} - 36 T^{5} + \cdots + 587584 \)
|
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