Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7774,2,Mod(1,7774)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7774, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("7774.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7774 = 2 \cdot 13^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7774.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: | \(62.0757025313\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.127049661.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 9x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 598) |
| 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
\( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 20x^{2} - 9x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 5\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 10\nu^{2} + 10\nu - 7 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 10\nu^{3} - 8\nu^{2} + 19\nu + 8 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + 9\beta_{3} + 11\beta_{2} + 29\beta _1 + 12 \)
|
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 |
|
1.00000 | −2.78499 | 1.00000 | −0.691569 | −2.78499 | −0.261325 | 1.00000 | 4.75618 | −0.691569 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.51059 | 1.00000 | 1.93177 | −1.51059 | −3.10973 | 1.00000 | −0.718104 | 1.93177 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.721424 | 1.00000 | 2.68859 | −0.721424 | 4.69405 | 1.00000 | −2.47955 | 2.68859 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.56823 | 1.00000 | −1.20377 | 1.56823 | −4.17415 | 1.00000 | −0.540645 | −1.20377 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 2.46215 | 1.00000 | 3.71462 | 2.46215 | 1.94789 | 1.00000 | 3.06218 | 3.71462 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.98663 | 1.00000 | 0.560360 | 2.98663 | −0.0967248 | 1.00000 | 5.91994 | 0.560360 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( +1 \) |
| \(23\) | \( +1 \) |
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 | 7774.2.a.z | 6 | |
| 13.b | even | 2 | 1 | 7774.2.a.w | 6 | ||
| 13.c | even | 3 | 2 | 598.2.e.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 598.2.e.a | ✓ | 12 | 13.c | even | 3 | 2 | |
| 7774.2.a.w | 6 | 13.b | even | 2 | 1 | ||
| 7774.2.a.z | 6 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 12T_{3}^{4} + 20T_{3}^{3} + 37T_{3}^{2} - 36T_{3} - 35 \)
|
|
\( T_{5}^{6} - 7T_{5}^{5} + 11T_{5}^{4} + 12T_{5}^{3} - 27T_{5}^{2} - 6T_{5} + 9 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots - 35 \)
|
| $5$ |
\( T^{6} - 7 T^{5} + \cdots + 9 \)
|
| $7$ |
\( T^{6} + T^{5} - 26 T^{4} + \cdots + 3 \)
|
| $11$ |
\( T^{6} - 2 T^{5} + \cdots - 9 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 831 \)
|
| $19$ |
\( T^{6} - 7 T^{5} + \cdots - 8165 \)
|
| $23$ |
\( (T + 1)^{6} \)
|
| $29$ |
\( T^{6} - T^{5} + \cdots - 2691 \)
|
| $31$ |
\( T^{6} - 9 T^{5} + \cdots + 163 \)
|
| $37$ |
\( T^{6} + T^{5} + \cdots - 729 \)
|
| $41$ |
\( T^{6} - 21 T^{5} + \cdots - 1521 \)
|
| $43$ |
\( T^{6} - 12 T^{5} + \cdots - 607 \)
|
| $47$ |
\( T^{6} - 9 T^{5} + \cdots - 1611 \)
|
| $53$ |
\( T^{6} - 10 T^{5} + \cdots + 16209 \)
|
| $59$ |
\( T^{6} - 13 T^{5} + \cdots + 81 \)
|
| $61$ |
\( T^{6} - 10 T^{5} + \cdots - 2653 \)
|
| $67$ |
\( T^{6} - 50 T^{4} + \cdots - 169 \)
|
| $71$ |
\( T^{6} - 5 T^{5} + \cdots + 29619 \)
|
| $73$ |
\( T^{6} + 2 T^{5} + \cdots - 135327 \)
|
| $79$ |
\( T^{6} - 35 T^{5} + \cdots - 60849 \)
|
| $83$ |
\( T^{6} - 38 T^{5} + \cdots + 84285 \)
|
| $89$ |
\( T^{6} + 5 T^{5} + \cdots - 5271 \)
|
| $97$ |
\( T^{6} - 8 T^{5} + \cdots + 19467 \)
|
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