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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 448) |
| 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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.
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 | −9.66025 | 0 | −11.8038 | 0 | 7.00000 | 0 | 66.3205 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 7.66025 | 0 | −22.1962 | 0 | 7.00000 | 0 | 31.6795 | 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.a | 2 | |
| 4.b | odd | 2 | 1 | 1792.4.a.c | 2 | ||
| 8.b | even | 2 | 1 | 1792.4.a.d | 2 | ||
| 8.d | odd | 2 | 1 | 1792.4.a.b | 2 | ||
| 16.e | even | 4 | 2 | 448.4.b.a | ✓ | 4 | |
| 16.f | odd | 4 | 2 | 448.4.b.b | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 448.4.b.a | ✓ | 4 | 16.e | even | 4 | 2 | |
| 448.4.b.b | yes | 4 | 16.f | odd | 4 | 2 | |
| 1792.4.a.a | 2 | 1.a | even | 1 | 1 | trivial | |
| 1792.4.a.b | 2 | 8.d | odd | 2 | 1 | ||
| 1792.4.a.c | 2 | 4.b | odd | 2 | 1 | ||
| 1792.4.a.d | 2 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1792))\):
|
\( T_{3}^{2} + 2T_{3} - 74 \)
|
|
\( T_{5}^{2} + 34T_{5} + 262 \)
|
|
\( T_{23}^{2} + 288T_{23} + 20148 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 2T - 74 \)
|
| $5$ |
\( T^{2} + 34T + 262 \)
|
| $7$ |
\( (T - 7)^{2} \)
|
| $11$ |
\( T^{2} - 4T - 1448 \)
|
| $13$ |
\( T^{2} + 74T + 862 \)
|
| $17$ |
\( T^{2} + 140T + 3172 \)
|
| $19$ |
\( T^{2} - 246T + 15102 \)
|
| $23$ |
\( T^{2} + 288T + 20148 \)
|
| $29$ |
\( T^{2} + 68T - 2312 \)
|
| $31$ |
\( T^{2} - 280T + 17872 \)
|
| $37$ |
\( T^{2} + 164T - 101576 \)
|
| $41$ |
\( T^{2} - 244T - 17564 \)
|
| $43$ |
\( T^{2} + 188T - 15464 \)
|
| $47$ |
\( T^{2} - 392T - 50336 \)
|
| $53$ |
\( T^{2} + 504T - 38064 \)
|
| $59$ |
\( T^{2} - 190T - 129650 \)
|
| $61$ |
\( T^{2} - 398T + 26134 \)
|
| $67$ |
\( T^{2} - 368T - 304832 \)
|
| $71$ |
\( T^{2} + 248T + 1504 \)
|
| $73$ |
\( T^{2} + 204T - 86796 \)
|
| $79$ |
\( T^{2} - 440T - 193568 \)
|
| $83$ |
\( T^{2} + 1158 T + 149238 \)
|
| $89$ |
\( T^{2} - 1156 T + 310852 \)
|
| $97$ |
\( T^{2} + 428T - 803276 \)
|
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