Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1792,4,Mod(1,1792)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, 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("1792.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1792 = 2^{8} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1792.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: | \(105.731422730\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.45177.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 11x^{2} + 12x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 896) |
| 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
\( x^{4} - 2x^{3} - 11x^{2} + 12x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 11\nu + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 5\nu + 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{3} + 4\nu^{2} + 24\nu - 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta _1 + 5 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} - \beta_{2} + 7\beta _1 + 55 ) / 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{3} + 5\beta_{2} + 13\beta _1 + 47 ) / 4 \)
|
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 | −8.34383 | 0 | −9.18125 | 0 | 7.00000 | 0 | 42.6195 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.99088 | 0 | 14.7739 | 0 | 7.00000 | 0 | −18.0546 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.738932 | 0 | −3.90152 | 0 | 7.00000 | 0 | −26.4540 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.07364 | 0 | −15.6912 | 0 | 7.00000 | 0 | 9.88912 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 1792.4.a.i | 4 | |
| 4.b | odd | 2 | 1 | 1792.4.a.m | 4 | ||
| 8.b | even | 2 | 1 | 1792.4.a.p | 4 | ||
| 8.d | odd | 2 | 1 | 1792.4.a.l | 4 | ||
| 16.e | even | 4 | 2 | 896.4.b.b | ✓ | 8 | |
| 16.f | odd | 4 | 2 | 896.4.b.d | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 896.4.b.b | ✓ | 8 | 16.e | even | 4 | 2 | |
| 896.4.b.d | yes | 8 | 16.f | odd | 4 | 2 | |
| 1792.4.a.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 1792.4.a.l | 4 | 8.d | odd | 2 | 1 | ||
| 1792.4.a.m | 4 | 4.b | odd | 2 | 1 | ||
| 1792.4.a.p | 4 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1792))\):
|
\( T_{3}^{4} + 6T_{3}^{3} - 40T_{3}^{2} - 184T_{3} - 112 \)
|
|
\( T_{5}^{4} + 14T_{5}^{3} - 184T_{5}^{2} - 3000T_{5} - 8304 \)
|
|
\( T_{23}^{4} - 36T_{23}^{3} - 37600T_{23}^{2} + 3797440T_{23} - 82329856 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 6 T^{3} + \cdots - 112 \)
|
| $5$ |
\( T^{4} + 14 T^{3} + \cdots - 8304 \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} + 56 T^{3} + \cdots + 1404928 \)
|
| $13$ |
\( T^{4} - 46 T^{3} + \cdots + 3507344 \)
|
| $17$ |
\( T^{4} - 136 T^{3} + \cdots - 153968 \)
|
| $19$ |
\( T^{4} + 58 T^{3} + \cdots + 2233616 \)
|
| $23$ |
\( T^{4} - 36 T^{3} + \cdots - 82329856 \)
|
| $29$ |
\( T^{4} + 244 T^{3} + \cdots + 45466624 \)
|
| $31$ |
\( T^{4} + 80 T^{3} + \cdots + 50577408 \)
|
| $37$ |
\( T^{4} - 116 T^{3} + \cdots + 852442112 \)
|
| $41$ |
\( T^{4} + \cdots - 1520856432 \)
|
| $43$ |
\( T^{4} + \cdots - 1507552256 \)
|
| $47$ |
\( T^{4} + \cdots + 1155252224 \)
|
| $53$ |
\( T^{4} + \cdots + 44581348096 \)
|
| $59$ |
\( T^{4} + \cdots + 115315103568 \)
|
| $61$ |
\( T^{4} + \cdots - 13967891568 \)
|
| $67$ |
\( T^{4} + \cdots - 10450884864 \)
|
| $71$ |
\( T^{4} + \cdots - 85775958016 \)
|
| $73$ |
\( T^{4} + \cdots + 20213706192 \)
|
| $79$ |
\( T^{4} + \cdots + 165281349632 \)
|
| $83$ |
\( T^{4} + \cdots - 1560931157296 \)
|
| $89$ |
\( T^{4} + \cdots + 571746550032 \)
|
| $97$ |
\( T^{4} + \cdots + 2676978896 \)
|
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