Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,4,Mod(1,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.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: | \(27.2588824227\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.1028796.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 295x + 175 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 295x + 175 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 4\nu - 195 ) / 10 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 36\nu + 185 ) / 10 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{2} + 9\beta _1 + 194 \)
|
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 |
|
−2.00000 | 3.00000 | 4.00000 | −17.2278 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | 34.4555 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | 3.00000 | 4.00000 | 4.54053 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −9.08106 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 3.00000 | 4.00000 | 19.6872 | −6.00000 | −7.00000 | −8.00000 | 9.00000 | −39.3745 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 462.4.a.r | ✓ | 3 |
| 3.b | odd | 2 | 1 | 1386.4.a.bh | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.4.a.r | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 1386.4.a.bh | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 7T_{5}^{2} - 328T_{5} + 1540 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(462))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T - 3)^{3} \)
|
| $5$ |
\( T^{3} - 7 T^{2} + \cdots + 1540 \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} + 15 T^{2} + \cdots + 54700 \)
|
| $17$ |
\( T^{3} + 40 T^{2} + \cdots + 205680 \)
|
| $19$ |
\( T^{3} - 41 T^{2} + \cdots - 129696 \)
|
| $23$ |
\( T^{3} - 266 T^{2} + \cdots - 568960 \)
|
| $29$ |
\( T^{3} - 39 T^{2} + \cdots + 2494196 \)
|
| $31$ |
\( T^{3} - 332 T^{2} + \cdots + 737600 \)
|
| $37$ |
\( T^{3} - 553 T^{2} + \cdots - 5036340 \)
|
| $41$ |
\( T^{3} + 320 T^{2} + \cdots - 1821936 \)
|
| $43$ |
\( T^{3} - 290 T^{2} + \cdots + 72543200 \)
|
| $47$ |
\( T^{3} - 33 T^{2} + \cdots - 5462672 \)
|
| $53$ |
\( T^{3} + 482 T^{2} + \cdots - 160561416 \)
|
| $59$ |
\( T^{3} + 443 T^{2} + \cdots - 4879248 \)
|
| $61$ |
\( T^{3} - 546 T^{2} + \cdots + 79584104 \)
|
| $67$ |
\( T^{3} - 661 T^{2} + \cdots + 24838512 \)
|
| $71$ |
\( T^{3} - 948 T^{2} + \cdots + 282521600 \)
|
| $73$ |
\( T^{3} - 1539 T^{2} + \cdots + 84284660 \)
|
| $79$ |
\( T^{3} - 1568 T^{2} + \cdots - 25989120 \)
|
| $83$ |
\( T^{3} - 52 T^{2} + \cdots + 106470528 \)
|
| $89$ |
\( T^{3} - 1042 T^{2} + \cdots + 597560136 \)
|
| $97$ |
\( T^{3} + \cdots - 1403321160 \)
|
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