Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [714,4,Mod(1,714)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, 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("714.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 714.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: | \(42.1273637441\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.356300.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 110x - 380 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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
\( x^{3} - 110x - 380 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 2\nu - 74 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 6\nu - 74 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 148 ) / 2 \)
|
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 | −19.3269 | 6.00000 | −7.00000 | −8.00000 | 9.00000 | 38.6539 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −3.00000 | 4.00000 | −0.213506 | 6.00000 | −7.00000 | −8.00000 | 9.00000 | 0.427013 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | −3.00000 | 4.00000 | 14.5404 | 6.00000 | −7.00000 | −8.00000 | 9.00000 | −29.0809 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( +1 \) |
| \(7\) | \( +1 \) |
| \(17\) | \( -1 \) |
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 | 714.4.a.j | ✓ | 3 |
| 3.b | odd | 2 | 1 | 2142.4.a.u | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 714.4.a.j | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 2142.4.a.u | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(714))\):
|
\( T_{5}^{3} + 5T_{5}^{2} - 280T_{5} - 60 \)
|
|
\( T_{11}^{3} + 35T_{11}^{2} - 1280T_{11} + 2480 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} + 5 T^{2} + \cdots - 60 \)
|
| $7$ |
\( (T + 7)^{3} \)
|
| $11$ |
\( T^{3} + 35 T^{2} + \cdots + 2480 \)
|
| $13$ |
\( T^{3} - 41 T^{2} + \cdots - 68 \)
|
| $17$ |
\( (T - 17)^{3} \)
|
| $19$ |
\( T^{3} - 79 T^{2} + \cdots + 343248 \)
|
| $23$ |
\( T^{3} + 99 T^{2} + \cdots - 60928 \)
|
| $29$ |
\( T^{3} + 10 T^{2} + \cdots - 290920 \)
|
| $31$ |
\( T^{3} + 204 T^{2} + \cdots - 8590848 \)
|
| $37$ |
\( T^{3} - 282 T^{2} + \cdots - 689464 \)
|
| $41$ |
\( T^{3} - 89 T^{2} + \cdots + 9795148 \)
|
| $43$ |
\( T^{3} - 131 T^{2} + \cdots - 71088 \)
|
| $47$ |
\( T^{3} - 236 T^{2} + \cdots - 150528 \)
|
| $53$ |
\( T^{3} - 356 T^{2} + \cdots - 1530848 \)
|
| $59$ |
\( T^{3} + 272 T^{2} + \cdots - 208129536 \)
|
| $61$ |
\( T^{3} - 562 T^{2} + \cdots - 4935384 \)
|
| $67$ |
\( T^{3} - 232 T^{2} + \cdots - 26838784 \)
|
| $71$ |
\( T^{3} + 2056 T^{2} + \cdots + 280593408 \)
|
| $73$ |
\( T^{3} + 254 T^{2} + \cdots + 148870152 \)
|
| $79$ |
\( T^{3} - 372 T^{2} + \cdots + 166027776 \)
|
| $83$ |
\( T^{3} + 618 T^{2} + \cdots + 68208416 \)
|
| $89$ |
\( T^{3} + 1612 T^{2} + \cdots + 25865664 \)
|
| $97$ |
\( T^{3} - 100 T^{2} + \cdots - 312950400 \)
|
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