Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2366,4,Mod(1,2366)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2366.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2366 = 2 \cdot 7 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2366.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: | \(139.598519074\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1169}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 292 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 182) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{1169})\). 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 |
|
2.00000 | −4.00000 | 4.00000 | −19.5953 | −8.00000 | −7.00000 | 8.00000 | −11.0000 | −39.1906 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | −4.00000 | 4.00000 | 14.5953 | −8.00000 | −7.00000 | 8.00000 | −11.0000 | 29.1906 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
| \(13\) | \(1\) |
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 | 2366.4.a.k | 2 | |
| 13.b | even | 2 | 1 | 182.4.a.f | ✓ | 2 | |
| 39.d | odd | 2 | 1 | 1638.4.a.p | 2 | ||
| 52.b | odd | 2 | 1 | 1456.4.a.l | 2 | ||
| 91.b | odd | 2 | 1 | 1274.4.a.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 182.4.a.f | ✓ | 2 | 13.b | even | 2 | 1 | |
| 1274.4.a.k | 2 | 91.b | odd | 2 | 1 | ||
| 1456.4.a.l | 2 | 52.b | odd | 2 | 1 | ||
| 1638.4.a.p | 2 | 39.d | odd | 2 | 1 | ||
| 2366.4.a.k | 2 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):
|
\( T_{3} + 4 \)
|
|
\( T_{5}^{2} + 5T_{5} - 286 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( (T + 4)^{2} \)
|
| $5$ |
\( T^{2} + 5T - 286 \)
|
| $7$ |
\( (T + 7)^{2} \)
|
| $11$ |
\( T^{2} - 22T - 1048 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 74T + 200 \)
|
| $19$ |
\( T^{2} + 41T - 13900 \)
|
| $23$ |
\( T^{2} - 105T + 2464 \)
|
| $29$ |
\( T^{2} - 341T + 28778 \)
|
| $31$ |
\( T^{2} + 81T - 22032 \)
|
| $37$ |
\( T^{2} + 378T + 34552 \)
|
| $41$ |
\( T^{2} + 672T + 108220 \)
|
| $43$ |
\( T^{2} + 163T + 4012 \)
|
| $47$ |
\( T^{2} - 61T - 127952 \)
|
| $53$ |
\( T^{2} + 7T - 128870 \)
|
| $59$ |
\( T^{2} - 484T - 170560 \)
|
| $61$ |
\( T^{2} - 108T - 296348 \)
|
| $67$ |
\( T^{2} - 518T + 65912 \)
|
| $71$ |
\( T^{2} - 1336 T + 277888 \)
|
| $73$ |
\( T^{2} + 1335 T + 438250 \)
|
| $79$ |
\( T^{2} + 1431 T + 427480 \)
|
| $83$ |
\( T^{2} + 1747 T + 657500 \)
|
| $89$ |
\( T^{2} - 193T - 348694 \)
|
| $97$ |
\( T^{2} + 1595T - 9574 \)
|
| show more | |
| show less |