Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1001,2,Mod(144,1001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 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("1001.144");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1001 = 7 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1001.i (of order \(3\), degree \(2\), 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: | \(7.99302524233\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.447703281.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} - 2x^{6} + 2x^{5} + 3x^{4} + 4x^{3} - 8x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 4\nu^{5} + 2\nu^{4} + \nu^{3} - 10\nu^{2} + 16 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + \nu^{5} - 2\nu^{3} - \nu^{2} + 6\nu + 8 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - \nu^{6} + 2\nu^{5} + 2\nu^{4} - \nu^{3} - 16\nu^{2} - 8\nu + 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - 6\nu^{4} + \nu^{3} + 16\nu^{2} + 4\nu - 24 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 4\nu^{5} + 3\nu^{3} + 8\nu^{2} - 2\nu - 20 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} + \nu^{6} - 6\nu^{5} - 6\nu^{4} + \nu^{3} + 24\nu^{2} + 12\nu - 32 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{7} + \nu^{6} - 9\nu^{5} - 6\nu^{4} + 6\nu^{3} + 27\nu^{2} + 4\nu - 40 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta _1 + 3 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{7} - \beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} + \beta_{2} + 4\beta_1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{6} - 4\beta_{5} - 3\beta_{4} - 4\beta_{3} - 5\beta_{2} + \beta _1 - 9 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 6\beta_{7} - 8\beta_{6} - 5\beta_{5} - 6\beta_{4} + \beta_{3} + 5\beta_{2} - 7\beta _1 - 12 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -3\beta_{7} - 11\beta_{6} + 4\beta_{5} + 15\beta_{4} - 11\beta_{3} + 2\beta_{2} + 5\beta _1 - 12 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1001\mathbb{Z}\right)^\times\).
| \(n\) | \(365\) | \(430\) | \(925\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 144.1 |
|
−1.18180 | − | 2.04695i | 0.0817820 | − | 0.141650i | −1.79332 | + | 3.10613i | −0.0817820 | − | 0.141650i | −0.386601 | −2.00000 | + | 1.73205i | 3.75021 | 1.48662 | + | 2.57491i | −0.193301 | + | 0.334806i | ||||||||||||||||||||||||||||
| 144.2 | −0.834713 | − | 1.44577i | −1.40601 | + | 2.43529i | −0.393492 | + | 0.681548i | 1.40601 | + | 2.43529i | 4.69447 | −2.00000 | + | 1.73205i | −2.02504 | −2.45374 | − | 4.25001i | 2.34723 | − | 4.06553i | |||||||||||||||||||||||||||||
| 144.3 | 0.0667052 | + | 0.115537i | 1.25673 | − | 2.17673i | 0.991101 | − | 1.71664i | −1.25673 | − | 2.17673i | 0.335323 | −2.00000 | + | 1.73205i | 0.531267 | −1.65876 | − | 2.87306i | 0.167661 | − | 0.290398i | |||||||||||||||||||||||||||||
| 144.4 | 0.949812 | + | 1.64512i | −0.432504 | + | 0.749119i | −0.804286 | + | 1.39306i | 0.432504 | + | 0.749119i | −1.64319 | −2.00000 | + | 1.73205i | 0.743565 | 1.12588 | + | 1.95008i | −0.821595 | + | 1.42304i | |||||||||||||||||||||||||||||
| 716.1 | −1.18180 | + | 2.04695i | 0.0817820 | + | 0.141650i | −1.79332 | − | 3.10613i | −0.0817820 | + | 0.141650i | −0.386601 | −2.00000 | − | 1.73205i | 3.75021 | 1.48662 | − | 2.57491i | −0.193301 | − | 0.334806i | |||||||||||||||||||||||||||||
| 716.2 | −0.834713 | + | 1.44577i | −1.40601 | − | 2.43529i | −0.393492 | − | 0.681548i | 1.40601 | − | 2.43529i | 4.69447 | −2.00000 | − | 1.73205i | −2.02504 | −2.45374 | + | 4.25001i | 2.34723 | + | 4.06553i | |||||||||||||||||||||||||||||
| 716.3 | 0.0667052 | − | 0.115537i | 1.25673 | + | 2.17673i | 0.991101 | + | 1.71664i | −1.25673 | + | 2.17673i | 0.335323 | −2.00000 | − | 1.73205i | 0.531267 | −1.65876 | + | 2.87306i | 0.167661 | + | 0.290398i | |||||||||||||||||||||||||||||
| 716.4 | 0.949812 | − | 1.64512i | −0.432504 | − | 0.749119i | −0.804286 | − | 1.39306i | 0.432504 | − | 0.749119i | −1.64319 | −2.00000 | − | 1.73205i | 0.743565 | 1.12588 | − | 1.95008i | −0.821595 | − | 1.42304i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1001.2.i.a | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 1001.2.i.a | ✓ | 8 |
| 7.c | even | 3 | 1 | 7007.2.a.m | 4 | ||
| 7.d | odd | 6 | 1 | 7007.2.a.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1001.2.i.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1001.2.i.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 7007.2.a.l | 4 | 7.d | odd | 6 | 1 | ||
| 7007.2.a.m | 4 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} + 2T_{2}^{7} + 8T_{2}^{6} + 6T_{2}^{5} + 29T_{2}^{4} + 24T_{2}^{3} + 53T_{2}^{2} - 7T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1001, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 1 \)
|
| $3$ |
\( T^{8} + T^{7} + 8 T^{6} + \cdots + 1 \)
|
| $5$ |
\( T^{8} - T^{7} + 8 T^{6} + \cdots + 1 \)
|
| $7$ |
\( (T^{2} + 4 T + 7)^{4} \)
|
| $11$ |
\( (T^{2} - T + 1)^{4} \)
|
| $13$ |
\( (T + 1)^{8} \)
|
| $17$ |
\( T^{8} + 9 T^{7} + \cdots + 441 \)
|
| $19$ |
\( T^{8} - 4 T^{7} + \cdots + 68121 \)
|
| $23$ |
\( T^{8} + 6 T^{7} + \cdots + 729 \)
|
| $29$ |
\( (T^{4} + 15 T^{3} + \cdots + 87)^{2} \)
|
| $31$ |
\( T^{8} - 10 T^{7} + \cdots + 1026169 \)
|
| $37$ |
\( T^{8} + 9 T^{7} + \cdots + 96721 \)
|
| $41$ |
\( (T^{4} + 5 T^{3} - 7 T^{2} + \cdots + 1)^{2} \)
|
| $43$ |
\( (T^{4} - 7 T^{3} + \cdots - 1391)^{2} \)
|
| $47$ |
\( T^{8} - 2 T^{7} + \cdots + 49 \)
|
| $53$ |
\( T^{8} - 27 T^{7} + \cdots + 19633761 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 12544 \)
|
| $61$ |
\( T^{8} + 19 T^{7} + \cdots + 178929 \)
|
| $67$ |
\( T^{8} - T^{7} + \cdots + 77841 \)
|
| $71$ |
\( (T^{4} + 5 T^{3} + \cdots + 625)^{2} \)
|
| $73$ |
\( T^{8} + 2 T^{7} + \cdots + 95481 \)
|
| $79$ |
\( T^{8} - 32 T^{7} + \cdots + 5257849 \)
|
| $83$ |
\( (T^{4} - 246 T^{2} + \cdots + 11421)^{2} \)
|
| $89$ |
\( T^{8} - 3 T^{7} + \cdots + 549081 \)
|
| $97$ |
\( (T^{4} - 3 T^{3} - 47 T^{2} + \cdots + 13)^{2} \)
|
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