Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,3,Mod(286,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("287.286");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 287.d (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(7.82018358714\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 120x^{6} + 2874x^{4} + 16920x^{2} + 19881 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{41}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 120x^{6} + 2874x^{4} + 16920x^{2} + 19881 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -83\nu^{6} - 10524\nu^{4} - 286059\nu^{2} - 1258848 ) / 191196 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{7} + 5\nu^{6} - 506\nu^{5} + 506\nu^{4} - 3795\nu^{3} + 3795\nu^{2} + 146076\nu - 50478 ) / 31866 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} - 5\nu^{6} + 506\nu^{5} - 506\nu^{4} + 3795\nu^{3} - 3795\nu^{2} - 82344\nu + 50478 ) / 31866 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 143\nu^{6} + 16596\nu^{4} + 331599\nu^{2} + 653112 ) / 191196 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2\nu^{6} + 225\nu^{4} + 4230\nu^{2} + 12285 ) / 1356 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -365\nu^{7} - 42249\nu^{5} - 882489\nu^{3} - 3564621\nu ) / 382392 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 59\nu^{7} + 7033\nu^{5} + 161623\nu^{3} + 675813\nu ) / 42488 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 8\beta_{5} - 21\beta_{4} - 9\beta _1 - 60 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -20\beta_{7} - 36\beta_{6} - 21\beta_{4} - 105\beta_{3} - 63\beta_{2} - 21\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -420\beta_{5} + 978\beta_{4} + 258\beta _1 + 2163 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2280\beta_{7} + 3960\beta_{6} + 1956\beta_{4} + 9291\beta_{3} + 5379\beta_{2} + 1956\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 78936\beta_{5} - 175635\beta_{4} - 39015\beta _1 - 372060 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -215556\beta_{7} - 373428\beta_{6} - 175635\beta_{4} - 831339\beta_{3} - 480069\beta_{2} - 175635\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).
| \(n\) | \(206\) | \(211\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 286.1 |
|
−1.00000 | −4.37780 | −3.00000 | − | 3.17602i | 4.37780 | 0.818350 | − | 6.95200i | 7.00000 | 10.1652 | 3.17602i | |||||||||||||||||||||||||||||||||||||||
| 286.2 | −1.00000 | −4.37780 | −3.00000 | 3.17602i | 4.37780 | 0.818350 | + | 6.95200i | 7.00000 | 10.1652 | − | 3.17602i | ||||||||||||||||||||||||||||||||||||||||
| 286.3 | −1.00000 | −0.913701 | −3.00000 | − | 7.47749i | 0.913701 | −6.10985 | − | 3.41609i | 7.00000 | −8.16515 | 7.47749i | ||||||||||||||||||||||||||||||||||||||||
| 286.4 | −1.00000 | −0.913701 | −3.00000 | 7.47749i | 0.913701 | −6.10985 | + | 3.41609i | 7.00000 | −8.16515 | − | 7.47749i | ||||||||||||||||||||||||||||||||||||||||
| 286.5 | −1.00000 | 0.913701 | −3.00000 | − | 7.47749i | −0.913701 | 6.10985 | + | 3.41609i | 7.00000 | −8.16515 | 7.47749i | ||||||||||||||||||||||||||||||||||||||||
| 286.6 | −1.00000 | 0.913701 | −3.00000 | 7.47749i | −0.913701 | 6.10985 | − | 3.41609i | 7.00000 | −8.16515 | − | 7.47749i | ||||||||||||||||||||||||||||||||||||||||
| 286.7 | −1.00000 | 4.37780 | −3.00000 | − | 3.17602i | −4.37780 | −0.818350 | + | 6.95200i | 7.00000 | 10.1652 | 3.17602i | ||||||||||||||||||||||||||||||||||||||||
| 286.8 | −1.00000 | 4.37780 | −3.00000 | 3.17602i | −4.37780 | −0.818350 | − | 6.95200i | 7.00000 | 10.1652 | − | 3.17602i | ||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 41.b | even | 2 | 1 | inner |
| 287.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 287.3.d.c | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 287.3.d.c | ✓ | 8 |
| 41.b | even | 2 | 1 | inner | 287.3.d.c | ✓ | 8 |
| 287.d | odd | 2 | 1 | inner | 287.3.d.c | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 287.3.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 287.3.d.c | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 287.3.d.c | ✓ | 8 | 41.b | even | 2 | 1 | inner |
| 287.3.d.c | ✓ | 8 | 287.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(287, [\chi])\):
|
\( T_{2} + 1 \)
|
|
\( T_{3}^{4} - 20T_{3}^{2} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( (T^{4} - 20 T^{2} + 16)^{2} \)
|
| $5$ |
\( (T^{4} + 66 T^{2} + 564)^{2} \)
|
| $7$ |
\( T^{8} + 44 T^{6} + \cdots + 5764801 \)
|
| $11$ |
\( (T^{4} + 414 T^{2} + 14100)^{2} \)
|
| $13$ |
\( (T^{4} - 20 T^{2} + 16)^{2} \)
|
| $17$ |
\( (T^{4} - 644 T^{2} + 784)^{2} \)
|
| $19$ |
\( (T^{4} - 1364 T^{2} + 446224)^{2} \)
|
| $23$ |
\( (T^{2} + 26 T - 20)^{4} \)
|
| $29$ |
\( (T^{4} + 2886 T^{2} + 352500)^{2} \)
|
| $31$ |
\( (T^{4} + 1032 T^{2} + 225600)^{2} \)
|
| $37$ |
\( (T^{2} + 56 T + 700)^{4} \)
|
| $41$ |
\( T^{8} + \cdots + 7984925229121 \)
|
| $43$ |
\( (T^{2} - 16 T - 692)^{4} \)
|
| $47$ |
\( (T^{4} - 5060 T^{2} + 6370576)^{2} \)
|
| $53$ |
\( (T^{4} + 6750 T^{2} + 9385524)^{2} \)
|
| $59$ |
\( (T^{4} + 810 T^{2} + 162996)^{2} \)
|
| $61$ |
\( (T^{4} + 6474 T^{2} + 4467444)^{2} \)
|
| $67$ |
\( (T^{4} + 3942 T^{2} + 772116)^{2} \)
|
| $71$ |
\( (T^{4} + 29808 T^{2} + 211242816)^{2} \)
|
| $73$ |
\( (T^{4} + 2376 T^{2} + 730944)^{2} \)
|
| $79$ |
\( (T^{4} + 7224 T^{2} + 11054400)^{2} \)
|
| $83$ |
\( (T^{4} + 35514 T^{2} + 223141524)^{2} \)
|
| $89$ |
\( (T^{4} - 11672 T^{2} + 33408400)^{2} \)
|
| $97$ |
\( (T^{2} - 10108)^{4} \)
|
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