Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2178,4,Mod(1,2178)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2178, 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("2178.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2178.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: | \(128.506159993\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). 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 | 0 | 4.00000 | −0.618034 | 0 | 9.79837 | 8.00000 | 0 | −1.23607 | ||||||||||||||||||||||||
| 1.2 | 2.00000 | 0 | 4.00000 | 1.61803 | 0 | −14.7984 | 8.00000 | 0 | 3.23607 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 2178.4.a.bi | 2 | |
| 3.b | odd | 2 | 1 | 242.4.a.h | 2 | ||
| 11.b | odd | 2 | 1 | 2178.4.a.z | 2 | ||
| 11.d | odd | 10 | 2 | 198.4.f.b | 4 | ||
| 12.b | even | 2 | 1 | 1936.4.a.bc | 2 | ||
| 33.d | even | 2 | 1 | 242.4.a.k | 2 | ||
| 33.f | even | 10 | 2 | 22.4.c.a | ✓ | 4 | |
| 33.f | even | 10 | 2 | 242.4.c.f | 4 | ||
| 33.h | odd | 10 | 2 | 242.4.c.j | 4 | ||
| 33.h | odd | 10 | 2 | 242.4.c.m | 4 | ||
| 132.d | odd | 2 | 1 | 1936.4.a.bb | 2 | ||
| 132.n | odd | 10 | 2 | 176.4.m.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.4.c.a | ✓ | 4 | 33.f | even | 10 | 2 | |
| 176.4.m.a | 4 | 132.n | odd | 10 | 2 | ||
| 198.4.f.b | 4 | 11.d | odd | 10 | 2 | ||
| 242.4.a.h | 2 | 3.b | odd | 2 | 1 | ||
| 242.4.a.k | 2 | 33.d | even | 2 | 1 | ||
| 242.4.c.f | 4 | 33.f | even | 10 | 2 | ||
| 242.4.c.j | 4 | 33.h | odd | 10 | 2 | ||
| 242.4.c.m | 4 | 33.h | odd | 10 | 2 | ||
| 1936.4.a.bb | 2 | 132.d | odd | 2 | 1 | ||
| 1936.4.a.bc | 2 | 12.b | even | 2 | 1 | ||
| 2178.4.a.z | 2 | 11.b | odd | 2 | 1 | ||
| 2178.4.a.bi | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2178))\):
|
\( T_{5}^{2} - T_{5} - 1 \)
|
|
\( T_{7}^{2} + 5T_{7} - 145 \)
|
|
\( T_{17}^{2} + 181T_{17} + 7279 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - T - 1 \)
|
| $7$ |
\( T^{2} + 5T - 145 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} - 37T - 709 \)
|
| $17$ |
\( T^{2} + 181T + 7279 \)
|
| $19$ |
\( T^{2} - 73T + 1231 \)
|
| $23$ |
\( T^{2} - 112T + 2416 \)
|
| $29$ |
\( T^{2} - 55T - 30845 \)
|
| $31$ |
\( T^{2} + 261T - 8531 \)
|
| $37$ |
\( T^{2} + 273T + 10831 \)
|
| $41$ |
\( T^{2} + 49T - 20201 \)
|
| $43$ |
\( T^{2} + 580T + 78320 \)
|
| $47$ |
\( T^{2} - 397T + 38851 \)
|
| $53$ |
\( T^{2} - 625T + 35495 \)
|
| $59$ |
\( T^{2} + 149T - 107701 \)
|
| $61$ |
\( T^{2} - 405T - 215505 \)
|
| $67$ |
\( T^{2} + 44T - 16336 \)
|
| $71$ |
\( T^{2} + 1125T + 73305 \)
|
| $73$ |
\( T^{2} + 51T - 2601 \)
|
| $79$ |
\( T^{2} + 1331T + 82429 \)
|
| $83$ |
\( T^{2} - 601T - 191731 \)
|
| $89$ |
\( T^{2} + 864T - 917876 \)
|
| $97$ |
\( T^{2} + 713T - 425689 \)
|
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