Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(75.7102679161\) |
| 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: |
\( x^{4} - x^{3} - 1058x^{2} - 2280x + 92160 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{3}\cdot 7 \) |
| 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
\( x^{4} - x^{3} - 1058x^{2} - 2280x + 92160 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu^{2} - 866\nu + 360 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 4\nu - 528 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta _1 + 528 \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} + 4\beta_{2} + 886\beta _1 + 2280 \)
|
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 |
|
−20.9958 | 81.0000 | −71.1763 | 1464.57 | −1700.66 | 0 | 12244.3 | 6561.00 | −30749.8 | ||||||||||||||||||||||||||||||
| 1.2 | −3.33717 | 81.0000 | −500.863 | −2645.54 | −270.311 | 0 | 3380.10 | 6561.00 | 8828.63 | |||||||||||||||||||||||||||||||
| 1.3 | 16.5523 | 81.0000 | −238.020 | 338.989 | 1340.74 | 0 | −12414.6 | 6561.00 | 5611.07 | |||||||||||||||||||||||||||||||
| 1.4 | 40.7806 | 81.0000 | 1151.06 | −1636.02 | 3303.23 | 0 | 26061.2 | 6561.00 | −66717.8 | |||||||||||||||||||||||||||||||
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 | 147.10.a.h | yes | 4 |
| 7.b | odd | 2 | 1 | 147.10.a.g | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.10.a.g | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 147.10.a.h | yes | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(147))\):
|
\( T_{2}^{4} - 33T_{2}^{3} - 650T_{2}^{2} + 12408T_{2} + 47296 \)
|
|
\( T_{5}^{4} + 2478T_{5}^{3} - 2897424T_{5}^{2} - 5680406880T_{5} + 2148816729600 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 33 T^{3} + \cdots + 47296 \)
|
| $3$ |
\( (T - 81)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 2148816729600 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 59\!\cdots\!60 \)
|
| $13$ |
\( T^{4} + \cdots - 73\!\cdots\!16 \)
|
| $17$ |
\( T^{4} + \cdots - 28\!\cdots\!32 \)
|
| $19$ |
\( T^{4} + \cdots + 33\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!28 \)
|
| $29$ |
\( T^{4} + \cdots - 26\!\cdots\!56 \)
|
| $31$ |
\( T^{4} + \cdots + 19\!\cdots\!92 \)
|
| $37$ |
\( T^{4} + \cdots + 23\!\cdots\!92 \)
|
| $41$ |
\( T^{4} + \cdots + 89\!\cdots\!80 \)
|
| $43$ |
\( T^{4} + \cdots - 41\!\cdots\!44 \)
|
| $47$ |
\( T^{4} + \cdots - 29\!\cdots\!92 \)
|
| $53$ |
\( T^{4} + \cdots - 34\!\cdots\!72 \)
|
| $59$ |
\( T^{4} + \cdots - 30\!\cdots\!72 \)
|
| $61$ |
\( T^{4} + \cdots - 26\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $71$ |
\( T^{4} + \cdots - 65\!\cdots\!48 \)
|
| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!40 \)
|
| $79$ |
\( T^{4} + \cdots + 26\!\cdots\!64 \)
|
| $83$ |
\( T^{4} + \cdots - 30\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots + 90\!\cdots\!28 \)
|
| $97$ |
\( T^{4} + \cdots - 25\!\cdots\!68 \)
|
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