Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1881,2,Mod(1,1881)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1881, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1881.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1881 = 3^{2} \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1881.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: | \(15.0198606202\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2179633.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 9x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 627) |
| 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
\( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 9x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 6\nu^{2} + 2\nu + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 7\beta_{2} + 2\beta _1 + 14 \)
|
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 |
|
−2.66695 | 0 | 5.11262 | −1.72144 | 0 | −3.18859 | −8.30122 | 0 | 4.59098 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.29410 | 0 | −0.325316 | −2.82255 | 0 | 2.68387 | 3.00918 | 0 | 3.65264 | ||||||||||||||||||||||||||||||||||
| 1.3 | −0.426019 | 0 | −1.81851 | 3.71870 | 0 | −0.191751 | 1.62676 | 0 | −1.58424 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.30941 | 0 | −0.285435 | −3.72141 | 0 | −3.27802 | −2.99258 | 0 | −4.87286 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.07765 | 0 | 2.31663 | 1.54669 | 0 | 2.97449 | 0.657857 | 0 | 3.21348 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
| \(19\) | \( -1 \) |
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 | 1881.2.a.l | 5 | |
| 3.b | odd | 2 | 1 | 627.2.a.i | ✓ | 5 | |
| 33.d | even | 2 | 1 | 6897.2.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 627.2.a.i | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 1881.2.a.l | 5 | 1.a | even | 1 | 1 | trivial | |
| 6897.2.a.r | 5 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1881))\):
|
\( T_{2}^{5} + T_{2}^{4} - 7T_{2}^{3} - 4T_{2}^{2} + 9T_{2} + 4 \)
|
|
\( T_{5}^{5} + 3T_{5}^{4} - 16T_{5}^{3} - 49T_{5}^{2} + 30T_{5} + 104 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + T^{4} - 7 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 3 T^{4} + \cdots + 104 \)
|
| $7$ |
\( T^{5} + T^{4} + \cdots + 16 \)
|
| $11$ |
\( (T - 1)^{5} \)
|
| $13$ |
\( T^{5} - 16 T^{4} + \cdots - 130 \)
|
| $17$ |
\( T^{5} + T^{4} + \cdots - 74 \)
|
| $19$ |
\( (T - 1)^{5} \)
|
| $23$ |
\( T^{5} - 5 T^{4} + \cdots + 128 \)
|
| $29$ |
\( T^{5} + 4 T^{4} + \cdots + 4696 \)
|
| $31$ |
\( T^{5} - 4 T^{4} + \cdots + 128 \)
|
| $37$ |
\( T^{5} - 7 T^{4} + \cdots - 22516 \)
|
| $41$ |
\( T^{5} + 7 T^{4} + \cdots - 404 \)
|
| $43$ |
\( T^{5} - 12 T^{4} + \cdots - 640 \)
|
| $47$ |
\( T^{5} + 2 T^{4} + \cdots + 16 \)
|
| $53$ |
\( T^{5} + 21 T^{4} + \cdots - 11542 \)
|
| $59$ |
\( T^{5} + 14 T^{4} + \cdots + 43420 \)
|
| $61$ |
\( T^{5} - 13 T^{4} + \cdots - 28124 \)
|
| $67$ |
\( T^{5} - 13 T^{4} + \cdots - 28936 \)
|
| $71$ |
\( T^{5} - 7 T^{4} + \cdots + 3904 \)
|
| $73$ |
\( T^{5} - 16 T^{4} + \cdots - 1976 \)
|
| $79$ |
\( T^{5} + 11 T^{4} + \cdots + 19456 \)
|
| $83$ |
\( T^{5} - 20 T^{4} + \cdots - 9140 \)
|
| $89$ |
\( T^{5} + 7 T^{4} + \cdots + 79582 \)
|
| $97$ |
\( T^{5} - 12 T^{4} + \cdots - 11164 \)
|
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