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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.9925933.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 6x^{4} + 6x^{3} + 10x^{2} + x - 1 \)
|
| 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
\( x^{6} - 2x^{5} - 6x^{4} + 6x^{3} + 10x^{2} + x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{5} + 3\nu^{4} + 3\nu^{3} - 9\nu^{2} - \nu + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 6\nu^{3} + 6\nu^{2} + 10\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( -\nu^{5} + 3\nu^{4} + 4\nu^{3} - 11\nu^{2} - 4\nu + 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{5} - 5\nu^{4} - 9\nu^{3} + 16\nu^{2} + 10\nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} + \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} + 5\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{5} + 3\beta_{4} - 8\beta_{3} + 7\beta_{2} + 9\beta _1 + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( 24\beta_{5} + 12\beta_{4} - 21\beta_{3} + 14\beta_{2} + 32\beta _1 + 43 \)
|
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 | −3.03724 | 1.00000 | 2.99731 | −3.03724 | 0.397449 | 1.00000 | 6.22486 | 2.99731 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −1.56319 | 1.00000 | −2.95405 | −1.56319 | 0.294010 | 1.00000 | −0.556445 | −2.95405 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −1.25879 | 1.00000 | −4.22940 | −1.25879 | −2.14521 | 1.00000 | −1.41546 | −4.22940 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 0.475026 | 1.00000 | 3.50582 | 0.475026 | −4.61358 | 1.00000 | −2.77435 | 3.50582 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.06162 | 1.00000 | −0.701244 | 1.06162 | −0.538465 | 1.00000 | −1.87297 | −0.701244 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.32258 | 1.00000 | −1.61845 | 2.32258 | 1.60580 | 1.00000 | 2.39436 | −1.61845 | |||||||||||||||||||||||||||||||||||||
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.y | 6 | |
| 13.b | even | 2 | 1 | 7774.2.a.v | 6 | ||
| 13.c | even | 3 | 2 | 598.2.e.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 598.2.e.b | ✓ | 12 | 13.c | even | 3 | 2 | |
| 7774.2.a.v | 6 | 13.b | even | 2 | 1 | ||
| 7774.2.a.y | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):
|
\( T_{3}^{6} + 2T_{3}^{5} - 8T_{3}^{4} - 12T_{3}^{3} + 13T_{3}^{2} + 12T_{3} - 7 \)
|
|
\( T_{5}^{6} + 3T_{5}^{5} - 21T_{5}^{4} - 60T_{5}^{3} + 91T_{5}^{2} + 298T_{5} + 149 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{6} \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots - 7 \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots + 149 \)
|
| $7$ |
\( T^{6} + 5 T^{5} + \cdots - 1 \)
|
| $11$ |
\( T^{6} + 12 T^{5} + \cdots + 123 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 2 T^{5} + \cdots - 1827 \)
|
| $19$ |
\( T^{6} - 3 T^{5} + \cdots - 49 \)
|
| $23$ |
\( (T - 1)^{6} \)
|
| $29$ |
\( T^{6} - 5 T^{5} + \cdots + 15909 \)
|
| $31$ |
\( T^{6} + 31 T^{5} + \cdots + 5119 \)
|
| $37$ |
\( T^{6} - 5 T^{5} + \cdots - 29513 \)
|
| $41$ |
\( T^{6} + 11 T^{5} + \cdots - 27593 \)
|
| $43$ |
\( T^{6} + 6 T^{5} + \cdots - 1475 \)
|
| $47$ |
\( T^{6} + 3 T^{5} + \cdots - 5051 \)
|
| $53$ |
\( T^{6} - 4 T^{5} + \cdots - 1087 \)
|
| $59$ |
\( T^{6} + 15 T^{5} + \cdots - 219 \)
|
| $61$ |
\( T^{6} - 6 T^{5} + \cdots - 21 \)
|
| $67$ |
\( T^{6} - 16 T^{5} + \cdots + 15503 \)
|
| $71$ |
\( T^{6} + 23 T^{5} + \cdots + 522739 \)
|
| $73$ |
\( T^{6} + 6 T^{5} + \cdots + 41869 \)
|
| $79$ |
\( T^{6} + 5 T^{5} + \cdots - 1061201 \)
|
| $83$ |
\( T^{6} + 30 T^{5} + \cdots + 43897 \)
|
| $89$ |
\( T^{6} + 29 T^{5} + \cdots + 542081 \)
|
| $97$ |
\( T^{6} + 34 T^{5} + \cdots + 523123 \)
|
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