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: | \(\Q(\sqrt{169009}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 42252 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 \(\beta = 9\sqrt{169009}\). 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 | −245886. | 0 | −823543. | 2.09715e6 | 0 | −3.14734e7 | ||||||||||||||||||||||||
| 1.2 | 128.000 | 0 | 16384.0 | 323908. | 0 | −823543. | 2.09715e6 | 0 | 4.14602e7 | |||||||||||||||||||||||||
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.k | 2 | |
| 3.b | odd | 2 | 1 | 14.16.a.b | ✓ | 2 | |
| 12.b | even | 2 | 1 | 112.16.a.b | 2 | ||
| 21.c | even | 2 | 1 | 98.16.a.c | 2 | ||
| 21.g | even | 6 | 2 | 98.16.c.g | 4 | ||
| 21.h | odd | 6 | 2 | 98.16.c.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.16.a.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 98.16.a.c | 2 | 21.c | even | 2 | 1 | ||
| 98.16.c.g | 4 | 21.g | even | 6 | 2 | ||
| 98.16.c.h | 4 | 21.h | odd | 6 | 2 | ||
| 112.16.a.b | 2 | 12.b | even | 2 | 1 | ||
| 126.16.a.k | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 78022T_{5} - 79644545120 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(126))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + \cdots - 79644545120 \)
|
| $7$ |
\( (T + 823543)^{2} \)
|
| $11$ |
\( T^{2} + \cdots - 15\!\cdots\!44 \)
|
| $13$ |
\( T^{2} + \cdots - 11\!\cdots\!68 \)
|
| $17$ |
\( T^{2} + \cdots - 11\!\cdots\!04 \)
|
| $19$ |
\( T^{2} + \cdots - 51\!\cdots\!40 \)
|
| $23$ |
\( T^{2} + \cdots + 23\!\cdots\!00 \)
|
| $29$ |
\( T^{2} + \cdots - 17\!\cdots\!80 \)
|
| $31$ |
\( T^{2} + \cdots - 20\!\cdots\!28 \)
|
| $37$ |
\( T^{2} + \cdots - 60\!\cdots\!24 \)
|
| $41$ |
\( T^{2} + \cdots - 23\!\cdots\!48 \)
|
| $43$ |
\( T^{2} + \cdots + 11\!\cdots\!48 \)
|
| $47$ |
\( T^{2} + \cdots + 45\!\cdots\!28 \)
|
| $53$ |
\( T^{2} + \cdots - 64\!\cdots\!72 \)
|
| $59$ |
\( T^{2} + \cdots - 15\!\cdots\!80 \)
|
| $61$ |
\( T^{2} + \cdots + 89\!\cdots\!40 \)
|
| $67$ |
\( T^{2} + \cdots - 31\!\cdots\!64 \)
|
| $71$ |
\( T^{2} + \cdots + 62\!\cdots\!52 \)
|
| $73$ |
\( T^{2} + \cdots - 32\!\cdots\!20 \)
|
| $79$ |
\( T^{2} + \cdots - 34\!\cdots\!80 \)
|
| $83$ |
\( T^{2} + \cdots + 60\!\cdots\!76 \)
|
| $89$ |
\( T^{2} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{2} + \cdots - 11\!\cdots\!20 \)
|
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