Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3864,2,Mod(1,3864)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, 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("3864.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3864.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: | \(30.8541953410\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 11x^{4} + 23x^{3} + 9x^{2} - 23x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 3x^{5} - 11x^{4} + 23x^{3} + 9x^{2} - 23x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 12\nu^{3} + 9\nu^{2} + 6\nu - 4 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 12\nu^{2} + 5\nu + 10 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} + 5\nu^{4} + 40\nu^{3} - 17\nu^{2} - 53\nu + 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} + 7\nu^{4} + 52\nu^{3} - 24\nu^{2} - 61\nu ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} + 9\nu^{4} + 66\nu^{3} - 37\nu^{2} - 87\nu + 16 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{3} - \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 4\beta_{4} - 6\beta_{3} - \beta_{2} + 3\beta _1 + 21 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} + \beta_{4} - 8\beta_{3} - 3\beta_{2} + 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 39\beta_{5} + 56\beta_{4} - 126\beta_{3} - 23\beta_{2} + 41\beta _1 + 263 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 255\beta_{5} + 124\beta_{4} - 570\beta_{3} - 175\beta_{2} + 69\beta _1 + 683 ) / 4 \)
|
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 | −2.60318 | 0 | −1.00000 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.00000 | 0 | −2.48633 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.00000 | 0 | −0.535873 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.00000 | 0 | 0.351005 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.00000 | 0 | 3.34309 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −1.00000 | 0 | 3.93128 | 0 | −1.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(23\) | \(1\) |
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 | 3864.2.a.w | ✓ | 6 |
| 4.b | odd | 2 | 1 | 7728.2.a.ch | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3864.2.a.w | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 7728.2.a.ch | 6 | 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3864))\):
|
\( T_{5}^{6} - 2T_{5}^{5} - 18T_{5}^{4} + 17T_{5}^{3} + 92T_{5}^{2} + 12T_{5} - 16 \)
|
|
\( T_{11}^{6} + 3T_{11}^{5} - 41T_{11}^{4} - 75T_{11}^{3} + 560T_{11}^{2} + 416T_{11} - 2368 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots - 16 \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots - 2368 \)
|
| $13$ |
\( T^{6} - 66 T^{4} + \cdots - 2264 \)
|
| $17$ |
\( T^{6} - 6 T^{5} + \cdots - 7168 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots - 2368 \)
|
| $23$ |
\( (T + 1)^{6} \)
|
| $29$ |
\( T^{6} - 5 T^{5} + \cdots - 5800 \)
|
| $31$ |
\( T^{6} + 4 T^{5} + \cdots - 12032 \)
|
| $37$ |
\( T^{6} - T^{5} + \cdots - 296 \)
|
| $41$ |
\( T^{6} - 12 T^{5} + \cdots + 67604 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + \cdots - 6784 \)
|
| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 4864 \)
|
| $53$ |
\( T^{6} - 10 T^{5} + \cdots - 800 \)
|
| $59$ |
\( T^{6} + 14 T^{5} + \cdots + 4256 \)
|
| $61$ |
\( T^{6} - 4 T^{5} + \cdots - 9208 \)
|
| $67$ |
\( T^{6} - 7 T^{5} + \cdots - 140288 \)
|
| $71$ |
\( T^{6} - 7 T^{5} + \cdots - 68096 \)
|
| $73$ |
\( T^{6} - 10 T^{5} + \cdots + 82304 \)
|
| $79$ |
\( T^{6} - 14 T^{5} + \cdots + 2048 \)
|
| $83$ |
\( T^{6} - 14 T^{5} + \cdots + 295808 \)
|
| $89$ |
\( T^{6} - 25 T^{5} + \cdots + 24016 \)
|
| $97$ |
\( T^{6} - 11 T^{5} + \cdots + 23896 \)
|
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