Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,8,Mod(1,147)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.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: | \(45.9205987462\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{4} - 2x^{3} - 299x^{2} - 1110x + 1890 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 7 \) |
| 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
\( x^{4} - 2x^{3} - 299x^{2} - 1110x + 1890 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{3} + 17\nu^{2} + 764\nu + 882 ) / 106 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 65\nu^{2} + 2728\nu + 9728 ) / 106 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - 82\nu^{2} - 262\nu + 9906 ) / 53 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} - 2\beta _1 + 20 ) / 42 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{3} + \beta_{2} - 7\beta _1 + 901 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -151\beta_{3} + 183\beta_{2} - 757\beta _1 + 17727 ) / 14 \)
|
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 |
|
−21.6612 | 27.0000 | 341.209 | 516.937 | −584.853 | 0 | −4618.37 | 729.000 | −11197.5 | ||||||||||||||||||||||||||||||
| 1.2 | −12.2117 | 27.0000 | 21.1267 | −284.670 | −329.717 | 0 | 1305.11 | 729.000 | 3476.32 | |||||||||||||||||||||||||||||||
| 1.3 | 0.702380 | 27.0000 | −127.507 | 168.328 | 18.9643 | 0 | −179.463 | 729.000 | 118.230 | |||||||||||||||||||||||||||||||
| 1.4 | 18.1706 | 27.0000 | 202.171 | 103.405 | 490.606 | 0 | 1347.73 | 729.000 | 1878.94 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 147.8.a.g | yes | 4 |
| 3.b | odd | 2 | 1 | 441.8.a.u | 4 | ||
| 7.b | odd | 2 | 1 | 147.8.a.f | ✓ | 4 | |
| 7.c | even | 3 | 2 | 147.8.e.l | 8 | ||
| 7.d | odd | 6 | 2 | 147.8.e.m | 8 | ||
| 21.c | even | 2 | 1 | 441.8.a.v | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.8.a.f | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 147.8.a.g | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 147.8.e.l | 8 | 7.c | even | 3 | 2 | ||
| 147.8.e.m | 8 | 7.d | odd | 6 | 2 | ||
| 441.8.a.u | 4 | 3.b | odd | 2 | 1 | ||
| 441.8.a.v | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(147))\):
|
\( T_{2}^{4} + 15T_{2}^{3} - 362T_{2}^{2} - 4560T_{2} + 3376 \)
|
|
\( T_{5}^{4} - 504T_{5}^{3} - 66636T_{5}^{2} + 35944560T_{5} - 2561414400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 15 T^{3} + \cdots + 3376 \)
|
| $3$ |
\( (T - 27)^{4} \)
|
| $5$ |
\( T^{4} + \cdots - 2561414400 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 25938734316800 \)
|
| $13$ |
\( T^{4} + \cdots - 48\!\cdots\!64 \)
|
| $17$ |
\( T^{4} + \cdots + 82\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots + 18\!\cdots\!16 \)
|
| $23$ |
\( T^{4} + \cdots - 98\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots + 22\!\cdots\!96 \)
|
| $31$ |
\( T^{4} + \cdots + 13\!\cdots\!56 \)
|
| $37$ |
\( T^{4} + \cdots - 29\!\cdots\!04 \)
|
| $41$ |
\( T^{4} + \cdots + 74\!\cdots\!00 \)
|
| $43$ |
\( T^{4} + \cdots + 47\!\cdots\!96 \)
|
| $47$ |
\( T^{4} + \cdots + 24\!\cdots\!96 \)
|
| $53$ |
\( T^{4} + \cdots - 53\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots - 89\!\cdots\!44 \)
|
| $61$ |
\( T^{4} + \cdots + 24\!\cdots\!36 \)
|
| $67$ |
\( T^{4} + \cdots + 28\!\cdots\!56 \)
|
| $71$ |
\( T^{4} + \cdots + 40\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 20\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 27\!\cdots\!04 \)
|
| $83$ |
\( T^{4} + \cdots + 39\!\cdots\!56 \)
|
| $89$ |
\( T^{4} + \cdots - 40\!\cdots\!84 \)
|
| $97$ |
\( T^{4} + \cdots + 13\!\cdots\!36 \)
|
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