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: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 3744535x^{2} - 1793673244x + 1139587195320 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{8}\cdot 5\cdot 7^{2} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 3744535x^{2} - 1793673244x + 1139587195320 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{3} + 5523\nu^{2} - 3461418\nu - 6286193004 ) / 79618 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -117\nu^{3} + 215397\nu^{2} + 226151946\nu - 245342100780 ) / 79618 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1266\nu^{3} - 1418718\nu^{2} - 2948743584\nu + 947491773900 ) / 39809 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 39\beta _1 + 2268 ) / 4536 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 198\beta_{3} + 4835\beta_{2} - 21453\beta _1 + 8492609916 ) / 4536 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 364518\beta_{3} + 7747429\beta_{2} - 114878955\beta _1 + 6127554201300 ) / 4536 \)
|
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 | −189588. | 0 | −823543. | 2.09715e6 | 0 | −2.42673e7 | ||||||||||||||||||||||||||||||
| 1.2 | 128.000 | 0 | 16384.0 | −134986. | 0 | −823543. | 2.09715e6 | 0 | −1.72783e7 | |||||||||||||||||||||||||||||||
| 1.3 | 128.000 | 0 | 16384.0 | 73616.2 | 0 | −823543. | 2.09715e6 | 0 | 9.42288e6 | |||||||||||||||||||||||||||||||
| 1.4 | 128.000 | 0 | 16384.0 | 195742. | 0 | −823543. | 2.09715e6 | 0 | 2.50550e7 | |||||||||||||||||||||||||||||||
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.p | yes | 4 |
| 3.b | odd | 2 | 1 | 126.16.a.o | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 126.16.a.o | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 126.16.a.p | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 55216T_{5}^{3} - 47425363680T_{5}^{2} - 2216321578788800T_{5} + 368773856677782646000 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(126))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 36\!\cdots\!00 \)
|
| $7$ |
\( (T + 823543)^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 63\!\cdots\!00 \)
|
| $13$ |
\( T^{4} + \cdots - 38\!\cdots\!08 \)
|
| $17$ |
\( T^{4} + \cdots + 46\!\cdots\!00 \)
|
| $19$ |
\( T^{4} + \cdots - 32\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots - 13\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 18\!\cdots\!08 \)
|
| $31$ |
\( T^{4} + \cdots - 15\!\cdots\!52 \)
|
| $37$ |
\( T^{4} + \cdots + 24\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 20\!\cdots\!28 \)
|
| $43$ |
\( T^{4} + \cdots + 38\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 22\!\cdots\!84 \)
|
| $53$ |
\( T^{4} + \cdots + 22\!\cdots\!96 \)
|
| $59$ |
\( T^{4} + \cdots - 53\!\cdots\!56 \)
|
| $61$ |
\( T^{4} + \cdots - 22\!\cdots\!72 \)
|
| $67$ |
\( T^{4} + \cdots - 46\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 34\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots + 45\!\cdots\!08 \)
|
| $79$ |
\( T^{4} + \cdots - 20\!\cdots\!08 \)
|
| $83$ |
\( T^{4} + \cdots - 77\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots + 34\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots + 77\!\cdots\!80 \)
|
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