Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2883,2,Mod(1,2883)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2883, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("2883.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2883 = 3 \cdot 31^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2883.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: | \(23.0208709027\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.1413480448.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 4x^{7} - 2x^{6} + 16x^{5} - x^{4} - 16x^{3} + 2x^{2} + 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{7} - 4\nu^{6} - 2\nu^{5} + 16\nu^{4} - \nu^{3} - 16\nu^{2} + 2\nu + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( -2\nu^{7} + 8\nu^{6} + 3\nu^{5} - 28\nu^{4} + 3\nu^{3} + 20\nu^{2} - 2\nu - 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\nu^{7} + 9\nu^{6} - \nu^{5} - 29\nu^{4} + 16\nu^{3} + 15\nu^{2} - 8\nu + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( -3\nu^{7} + 12\nu^{6} + 5\nu^{5} - 44\nu^{4} + 4\nu^{3} + 35\nu^{2} - 2\nu - 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\nu^{7} - 7\nu^{6} - 8\nu^{5} + 30\nu^{4} + 14\nu^{3} - 33\nu^{2} - 11\nu + 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( -2\nu^{7} + 6\nu^{6} + 12\nu^{5} - 29\nu^{4} - 26\nu^{3} + 35\nu^{2} + 16\nu - 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{3} - \beta_{2} + 2\beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + 2\beta_{3} - 3\beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{7} + 2\beta_{6} - 13\beta_{5} + 4\beta_{4} + 9\beta_{3} - 11\beta_{2} + 20\beta _1 + 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( 13\beta_{7} + 9\beta_{6} - 43\beta_{5} + 17\beta_{4} + 25\beta_{3} - 37\beta_{2} + 65\beta _1 - 1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 42\beta_{7} + 25\beta_{6} - 151\beta_{5} + 60\beta_{4} + 87\beta_{3} - 125\beta_{2} + 205\beta _1 - 1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 147\beta_{7} + 87\beta_{6} - 501\beta_{5} + 211\beta_{4} + 272\beta_{3} - 416\beta_{2} + 667\beta _1 - 25 \)
|
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 |
|
−2.28192 | 1.00000 | 3.20718 | 2.11277 | −2.28192 | 1.56903 | −2.75468 | 1.00000 | −4.82118 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −0.987714 | 1.00000 | −1.02442 | 4.03512 | −0.987714 | −3.85422 | 2.98726 | 1.00000 | −3.98554 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −0.137609 | 1.00000 | −1.98106 | −0.677863 | −0.137609 | 2.14378 | 0.547830 | 1.00000 | 0.0932799 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.573500 | 1.00000 | −1.67110 | −1.97206 | 0.573500 | 0.440009 | −2.10538 | 1.00000 | −1.13098 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.680299 | 1.00000 | −1.53719 | 0.966483 | 0.680299 | −4.84476 | −2.40635 | 1.00000 | 0.657497 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.73391 | 1.00000 | 1.00646 | −1.94715 | 1.73391 | 4.25897 | −1.72271 | 1.00000 | −3.37619 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.86771 | 1.00000 | 1.48834 | 1.48102 | 1.86771 | −4.98324 | −0.955629 | 1.00000 | 2.76613 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.55182 | 1.00000 | 4.51180 | −3.99832 | 2.55182 | −2.72957 | 6.40966 | 1.00000 | −10.2030 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(31\) | \( -1 \) |
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 | 2883.2.a.r | yes | 8 |
| 3.b | odd | 2 | 1 | 8649.2.a.bd | 8 | ||
| 31.b | odd | 2 | 1 | 2883.2.a.q | ✓ | 8 | |
| 93.c | even | 2 | 1 | 8649.2.a.bc | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2883.2.a.q | ✓ | 8 | 31.b | odd | 2 | 1 | |
| 2883.2.a.r | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 8649.2.a.bc | 8 | 93.c | even | 2 | 1 | ||
| 8649.2.a.bd | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2883))\):
|
\( T_{2}^{8} - 4T_{2}^{7} - 2T_{2}^{6} + 24T_{2}^{5} - 21T_{2}^{4} - 16T_{2}^{3} + 22T_{2}^{2} - 4T_{2} - 1 \)
|
|
\( T_{11}^{8} + 8T_{11}^{7} - 4T_{11}^{6} - 184T_{11}^{5} - 506T_{11}^{4} - 112T_{11}^{3} + 1328T_{11}^{2} + 1872T_{11} + 764 \)
|
|
\( T_{13}^{8} + 16T_{13}^{7} + 68T_{13}^{6} - 176T_{13}^{5} - 2226T_{13}^{4} - 6240T_{13}^{3} - 5112T_{13}^{2} + 4000T_{13} + 5956 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 4 T^{7} + \cdots - 1 \)
|
| $3$ |
\( (T - 1)^{8} \)
|
| $5$ |
\( T^{8} - 24 T^{6} + \cdots + 127 \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots + 1601 \)
|
| $11$ |
\( T^{8} + 8 T^{7} + \cdots + 764 \)
|
| $13$ |
\( T^{8} + 16 T^{7} + \cdots + 5956 \)
|
| $17$ |
\( T^{8} - 56 T^{6} + \cdots - 64 \)
|
| $19$ |
\( T^{8} + 16 T^{7} + \cdots + 10369 \)
|
| $23$ |
\( T^{8} + 16 T^{7} + \cdots - 772 \)
|
| $29$ |
\( T^{8} + 16 T^{7} + \cdots + 7996 \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} + 16 T^{7} + \cdots + 4 \)
|
| $41$ |
\( T^{8} - 244 T^{6} + \cdots + 2938879 \)
|
| $43$ |
\( T^{8} + 32 T^{7} + \cdots + 4 \)
|
| $47$ |
\( T^{8} + 8 T^{7} + \cdots - 69376 \)
|
| $53$ |
\( T^{8} + 16 T^{7} + \cdots - 392192 \)
|
| $59$ |
\( T^{8} - 24 T^{7} + \cdots + 813343 \)
|
| $61$ |
\( T^{8} + 48 T^{7} + \cdots + 43072 \)
|
| $67$ |
\( T^{8} + 40 T^{7} + \cdots + 10211584 \)
|
| $71$ |
\( T^{8} - 16 T^{7} + \cdots - 92993 \)
|
| $73$ |
\( T^{8} + 48 T^{7} + \cdots + 1604 \)
|
| $79$ |
\( T^{8} - 296 T^{6} + \cdots + 12356 \)
|
| $83$ |
\( T^{8} + 24 T^{7} + \cdots - 31610948 \)
|
| $89$ |
\( T^{8} + 24 T^{7} + \cdots - 23670788 \)
|
| $97$ |
\( T^{8} - 252 T^{6} + \cdots + 3697 \)
|
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