Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,4,Mod(1,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, 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("392.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.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: | \(23.1287487223\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{113})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 67\nu - 34 ) / 105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{3} + 6\nu^{2} + 344\nu - 173 ) / 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 51\nu^{2} - 86\nu - 1558 ) / 105 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 2\beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + \beta_{2} + 61 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{3} + 35\beta_{2} - 172\beta _1 + 91 ) / 2 \)
|
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 |
|
0 | −9.63797 | 0 | 5.91838 | 0 | 0 | 0 | 65.8904 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −5.39533 | 0 | −20.9517 | 0 | 0 | 0 | 2.10956 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 5.39533 | 0 | 20.9517 | 0 | 0 | 0 | 2.10956 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.63797 | 0 | −5.91838 | 0 | 0 | 0 | 65.8904 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.4.a.n | ✓ | 4 |
| 4.b | odd | 2 | 1 | 784.4.a.bh | 4 | ||
| 7.b | odd | 2 | 1 | inner | 392.4.a.n | ✓ | 4 |
| 7.c | even | 3 | 2 | 392.4.i.o | 8 | ||
| 7.d | odd | 6 | 2 | 392.4.i.o | 8 | ||
| 28.d | even | 2 | 1 | 784.4.a.bh | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 392.4.a.n | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 392.4.a.n | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 392.4.i.o | 8 | 7.c | even | 3 | 2 | ||
| 392.4.i.o | 8 | 7.d | odd | 6 | 2 | ||
| 784.4.a.bh | 4 | 4.b | odd | 2 | 1 | ||
| 784.4.a.bh | 4 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\):
|
\( T_{3}^{4} - 122T_{3}^{2} + 2704 \)
|
|
\( T_{5}^{4} - 474T_{5}^{2} + 15376 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 122T^{2} + 2704 \)
|
| $5$ |
\( T^{4} - 474 T^{2} + 15376 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 58 T + 728)^{2} \)
|
| $13$ |
\( T^{4} - 10242 T^{2} + 16257024 \)
|
| $17$ |
\( (T^{2} - 8450)^{2} \)
|
| $19$ |
\( T^{4} - 32682 T^{2} + 266211856 \)
|
| $23$ |
\( (T^{2} + 40 T - 6832)^{2} \)
|
| $29$ |
\( (T^{2} - 18 T - 5456)^{2} \)
|
| $31$ |
\( T^{4} - 32968 T^{2} + 154157056 \)
|
| $37$ |
\( (T^{2} - 218 T - 28912)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 4262261796 \)
|
| $43$ |
\( (T^{2} - 186 T + 8536)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 6407682304 \)
|
| $53$ |
\( (T^{2} - 488 T + 22924)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 242288403984 \)
|
| $61$ |
\( T^{4} - 86250 T^{2} + 756250000 \)
|
| $67$ |
\( (T^{2} + 588 T + 31744)^{2} \)
|
| $71$ |
\( (T^{2} + 1204 T + 163072)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 5359118436 \)
|
| $79$ |
\( (T^{2} - 28 T - 76192)^{2} \)
|
| $83$ |
\( T^{4} - 2714 T^{2} + 1547536 \)
|
| $89$ |
\( T^{4} + \cdots + 62037857476 \)
|
| $97$ |
\( T^{4} + \cdots + 9932514244 \)
|
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