Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2088,4,Mod(1,2088)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2088, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("2088.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2088.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: | \(123.195988092\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 232) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 34x^{3} + 74x^{2} + 94x - 198 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 64\nu - 90 ) / 19 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} - 5\nu^{3} + 23\nu^{2} + 140\nu - 109 ) / 19 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} - 6\nu^{3} + 134\nu^{2} - 60\nu - 203 ) / 19 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -9\nu^{4} - 7\nu^{3} + 283\nu^{2} - 184\nu - 829 ) / 19 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} - 2\beta_{3} + \beta _1 + 27 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{4} + 4\beta_{3} + 10\beta_{2} - 21\beta _1 - 43 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{4} - 33\beta_{3} - 9\beta_{2} + 29\beta _1 + 344 \)
|
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 | 0 | 0 | −16.3850 | 0 | 5.74609 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −15.2584 | 0 | 33.3511 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | −0.397400 | 0 | −14.4223 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0 | 0 | 7.04208 | 0 | 14.1473 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 0 | 0 | 14.9988 | 0 | −6.82211 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(29\) | \(1\) |
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 | 2088.4.a.f | 5 | |
| 3.b | odd | 2 | 1 | 232.4.a.d | ✓ | 5 | |
| 12.b | even | 2 | 1 | 464.4.a.m | 5 | ||
| 24.f | even | 2 | 1 | 1856.4.a.ba | 5 | ||
| 24.h | odd | 2 | 1 | 1856.4.a.z | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 232.4.a.d | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 464.4.a.m | 5 | 12.b | even | 2 | 1 | ||
| 1856.4.a.z | 5 | 24.h | odd | 2 | 1 | ||
| 1856.4.a.ba | 5 | 24.f | even | 2 | 1 | ||
| 2088.4.a.f | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} + 10T_{5}^{4} - 338T_{5}^{3} - 2304T_{5}^{2} + 25545T_{5} + 10494 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2088))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 10 T^{4} + \cdots + 10494 \)
|
| $7$ |
\( T^{5} - 32 T^{4} + \cdots - 266752 \)
|
| $11$ |
\( T^{5} + 36 T^{4} + \cdots - 1741860 \)
|
| $13$ |
\( T^{5} - 26 T^{4} + \cdots + 105404410 \)
|
| $17$ |
\( T^{5} + 82 T^{4} + \cdots + 49184 \)
|
| $19$ |
\( T^{5} - 156 T^{4} + \cdots - 1820736 \)
|
| $23$ |
\( T^{5} + \cdots - 7489438848 \)
|
| $29$ |
\( (T + 29)^{5} \)
|
| $31$ |
\( T^{5} - 432 T^{4} + \cdots + 445071048 \)
|
| $37$ |
\( T^{5} + \cdots - 15294686720 \)
|
| $41$ |
\( T^{5} + \cdots + 731491061376 \)
|
| $43$ |
\( T^{5} + \cdots - 571309913052 \)
|
| $47$ |
\( T^{5} + \cdots - 2559413417896 \)
|
| $53$ |
\( T^{5} + \cdots - 103910584482 \)
|
| $59$ |
\( T^{5} + \cdots - 16799541984192 \)
|
| $61$ |
\( T^{5} + \cdots + 2366067286944 \)
|
| $67$ |
\( T^{5} + \cdots + 29804817076224 \)
|
| $71$ |
\( T^{5} + \cdots + 50004302698368 \)
|
| $73$ |
\( T^{5} + \cdots - 4755790305792 \)
|
| $79$ |
\( T^{5} + \cdots - 13404287016 \)
|
| $83$ |
\( T^{5} + \cdots + 181825403631808 \)
|
| $89$ |
\( T^{5} + \cdots + 79946879886976 \)
|
| $97$ |
\( T^{5} + \cdots - 41073987679360 \)
|
| show more | |
| show less |