Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1407,2,Mod(1,1407)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1407.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1407 = 3 \cdot 7 \cdot 67 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1407.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: | \(11.2349515644\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.3512000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 7x^{4} + 4x^{3} + 11x^{2} - 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 7x^{4} + 4x^{3} + 11x^{2} - 3x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 4\nu^{2} + \nu + 1 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 6\nu^{3} - 3\nu^{2} - 5\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 6\nu^{2} + 3\nu + 5 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 6\nu^{3} + 3\nu^{2} + 9\nu - 2 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 15\nu - 4 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{5} + 3\beta_{4} - \beta_{3} + 2\beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{5} + 9\beta_{4} - 10\beta_{3} + 7\beta_{2} + 14\beta _1 + 23 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -14\beta_{5} + 37\beta_{4} - 16\beta_{3} + 19\beta_{2} + 20\beta _1 + 39 ) / 2 \)
|
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 |
|
−2.16680 | −1.00000 | 2.69501 | 3.23607 | 2.16680 | 1.00000 | −1.50595 | 1.00000 | −7.01191 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1.39627 | −1.00000 | −0.0504168 | −1.23607 | 1.39627 | 1.00000 | 2.86295 | 1.00000 | 1.72589 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.687423 | −1.00000 | −1.52745 | −1.23607 | 0.687423 | 1.00000 | 2.42485 | 1.00000 | 0.849702 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −0.364625 | −1.00000 | −1.86705 | 3.23607 | 0.364625 | 1.00000 | 1.41002 | 1.00000 | −1.17995 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.08370 | −1.00000 | 2.34180 | −1.23607 | −2.08370 | 1.00000 | 0.712204 | 1.00000 | −2.57559 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.53142 | −1.00000 | 4.40810 | 3.23607 | −2.53142 | 1.00000 | 6.09593 | 1.00000 | 8.19186 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(67\) | \(1\) |
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 | 1407.2.a.j | ✓ | 6 |
| 3.b | odd | 2 | 1 | 4221.2.a.u | 6 | ||
| 7.b | odd | 2 | 1 | 9849.2.a.bc | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1407.2.a.j | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 4221.2.a.u | 6 | 3.b | odd | 2 | 1 | ||
| 9849.2.a.bc | 6 | 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1407))\):
|
\( T_{2}^{6} - 9T_{2}^{4} - 4T_{2}^{3} + 19T_{2}^{2} + 18T_{2} + 4 \)
|
|
\( T_{5}^{2} - 2T_{5} - 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 9 T^{4} + \cdots + 4 \)
|
| $3$ |
\( (T + 1)^{6} \)
|
| $5$ |
\( (T^{2} - 2 T - 4)^{3} \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots + 116 \)
|
| $13$ |
\( T^{6} - 16 T^{5} + \cdots - 304 \)
|
| $17$ |
\( T^{6} - 14 T^{5} + \cdots - 16 \)
|
| $19$ |
\( T^{6} + 8 T^{5} + \cdots + 176 \)
|
| $23$ |
\( T^{6} + 8 T^{5} + \cdots - 1616 \)
|
| $29$ |
\( T^{6} - 12 T^{5} + \cdots - 13744 \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots - 2804 \)
|
| $37$ |
\( T^{6} + 10 T^{5} + \cdots + 1661 \)
|
| $41$ |
\( T^{6} - 32 T^{5} + \cdots - 2816 \)
|
| $43$ |
\( T^{6} + 18 T^{5} + \cdots - 704 \)
|
| $47$ |
\( T^{6} - 6 T^{5} + \cdots - 66821 \)
|
| $53$ |
\( T^{6} - 20 T^{5} + \cdots + 1423936 \)
|
| $59$ |
\( T^{6} - 6 T^{5} + \cdots + 6691 \)
|
| $61$ |
\( T^{6} + 14 T^{5} + \cdots - 1276 \)
|
| $67$ |
\( (T + 1)^{6} \)
|
| $71$ |
\( T^{6} - 12 T^{5} + \cdots + 27904 \)
|
| $73$ |
\( T^{6} - 12 T^{5} + \cdots + 62864 \)
|
| $79$ |
\( T^{6} + 14 T^{5} + \cdots + 64 \)
|
| $83$ |
\( T^{6} - 4 T^{5} + \cdots - 29696 \)
|
| $89$ |
\( T^{6} - 6 T^{5} + \cdots - 40189 \)
|
| $97$ |
\( T^{6} - 28 T^{5} + \cdots - 141136 \)
|
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