Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [126,16,Mod(1,126)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(126, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("126.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 126.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: | \(179.793816426\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{2} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 2928912 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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 \(\beta = 42\sqrt{11715649}\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
128.000 | 0 | 16384.0 | −98462.1 | 0 | −823543. | 2.09715e6 | 0 | −1.26032e7 | ||||||||||||||||||||||||
| 1.2 | 128.000 | 0 | 16384.0 | 189054. | 0 | −823543. | 2.09715e6 | 0 | 2.41989e7 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(7\) | \( +1 \) |
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 | 126.16.a.l | 2 | |
| 3.b | odd | 2 | 1 | 42.16.a.f | ✓ | 2 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.16.a.f | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 126.16.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 90592T_{5} - 18614677220 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(126))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + \cdots - 18614677220 \)
|
| $7$ |
\( (T + 823543)^{2} \)
|
| $11$ |
\( T^{2} + \cdots + 751888445030272 \)
|
| $13$ |
\( T^{2} + \cdots - 80\!\cdots\!36 \)
|
| $17$ |
\( T^{2} + \cdots - 62\!\cdots\!76 \)
|
| $19$ |
\( T^{2} + \cdots - 31\!\cdots\!92 \)
|
| $23$ |
\( T^{2} + \cdots - 23\!\cdots\!60 \)
|
| $29$ |
\( T^{2} + \cdots - 37\!\cdots\!36 \)
|
| $31$ |
\( T^{2} + \cdots - 94\!\cdots\!28 \)
|
| $37$ |
\( T^{2} + \cdots - 39\!\cdots\!84 \)
|
| $41$ |
\( T^{2} + \cdots - 23\!\cdots\!28 \)
|
| $43$ |
\( T^{2} + \cdots - 10\!\cdots\!44 \)
|
| $47$ |
\( T^{2} + \cdots - 29\!\cdots\!68 \)
|
| $53$ |
\( T^{2} + \cdots + 31\!\cdots\!32 \)
|
| $59$ |
\( T^{2} + \cdots + 10\!\cdots\!08 \)
|
| $61$ |
\( T^{2} + \cdots - 59\!\cdots\!80 \)
|
| $67$ |
\( T^{2} + \cdots + 16\!\cdots\!64 \)
|
| $71$ |
\( T^{2} + \cdots + 78\!\cdots\!08 \)
|
| $73$ |
\( T^{2} + \cdots + 14\!\cdots\!00 \)
|
| $79$ |
\( T^{2} + \cdots - 15\!\cdots\!00 \)
|
| $83$ |
\( T^{2} + \cdots - 12\!\cdots\!32 \)
|
| $89$ |
\( T^{2} + \cdots - 10\!\cdots\!80 \)
|
| $97$ |
\( T^{2} + \cdots - 28\!\cdots\!00 \)
|
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