Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1386,2,Mod(793,1386)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1386, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1386.793");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1386.k (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: | \(11.0672657201\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.21870000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 462) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} + 24x^{4} - 43x^{3} + 138x^{2} - 117x + 73 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} + 5\nu^{4} - 30\nu^{3} + 40\nu^{2} - 70\nu + 13 ) / 31 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} + 18\nu^{4} - 15\nu^{3} + 144\nu^{2} + 151\nu - 164 ) / 62 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} - 13\nu^{4} + 47\nu^{3} - 197\nu^{2} + 461\nu - 133 ) / 62 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 13\nu^{4} + 16\nu^{3} - 166\nu^{2} + 151\nu - 102 ) / 31 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -2\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} + 2\beta _1 - 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{5} + 4\beta_{4} - 2\beta_{3} + \beta_{2} - 8\beta _1 - 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{5} - 16\beta_{4} + 20\beta_{3} - 9\beta_{2} - 28\beta _1 + 75 \)
|
| \(\nu^{5}\) | \(=\) |
\( 45\beta_{5} - 60\beta_{4} + 40\beta_{3} - 33\beta_{2} + 55\beta _1 + 139 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(1135\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 793.1 |
|
0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.40280 | + | 2.42972i | 0 | −1.40280 | − | 2.24325i | −1.00000 | 0 | 1.40280 | + | 2.42972i | ||||||||||||||||||||||||||
| 793.2 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.806615 | + | 1.39710i | 0 | −0.806615 | + | 2.51980i | −1.00000 | 0 | 0.806615 | + | 1.39710i | |||||||||||||||||||||||||||
| 793.3 | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 0.866025i | 2.20942 | − | 3.82682i | 0 | 2.20942 | + | 1.45550i | −1.00000 | 0 | −2.20942 | − | 3.82682i | |||||||||||||||||||||||||||
| 991.1 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.40280 | − | 2.42972i | 0 | −1.40280 | + | 2.24325i | −1.00000 | 0 | 1.40280 | − | 2.42972i | |||||||||||||||||||||||||||
| 991.2 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.806615 | − | 1.39710i | 0 | −0.806615 | − | 2.51980i | −1.00000 | 0 | 0.806615 | − | 1.39710i | |||||||||||||||||||||||||||
| 991.3 | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 0.866025i | 2.20942 | + | 3.82682i | 0 | 2.20942 | − | 1.45550i | −1.00000 | 0 | −2.20942 | + | 3.82682i | |||||||||||||||||||||||||||
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 | 1386.2.k.v | 6 | |
| 3.b | odd | 2 | 1 | 462.2.i.g | ✓ | 6 | |
| 7.c | even | 3 | 1 | inner | 1386.2.k.v | 6 | |
| 7.c | even | 3 | 1 | 9702.2.a.dv | 3 | ||
| 7.d | odd | 6 | 1 | 9702.2.a.dw | 3 | ||
| 21.g | even | 6 | 1 | 3234.2.a.bh | 3 | ||
| 21.h | odd | 6 | 1 | 462.2.i.g | ✓ | 6 | |
| 21.h | odd | 6 | 1 | 3234.2.a.bf | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.2.i.g | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 462.2.i.g | ✓ | 6 | 21.h | odd | 6 | 1 | |
| 1386.2.k.v | 6 | 1.a | even | 1 | 1 | trivial | |
| 1386.2.k.v | 6 | 7.c | even | 3 | 1 | inner | |
| 3234.2.a.bf | 3 | 21.h | odd | 6 | 1 | ||
| 3234.2.a.bh | 3 | 21.g | even | 6 | 1 | ||
| 9702.2.a.dv | 3 | 7.c | even | 3 | 1 | ||
| 9702.2.a.dw | 3 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\):
|
\( T_{5}^{6} + 15T_{5}^{4} + 40T_{5}^{3} + 225T_{5}^{2} + 300T_{5} + 400 \)
|
|
\( T_{13} \)
|
|
\( T_{17}^{6} - 3T_{17}^{5} + 66T_{17}^{4} - 107T_{17}^{3} + 3666T_{17}^{2} - 7923T_{17} + 19321 \)
|
|
\( T_{23}^{6} - 3T_{23}^{5} + 36T_{23}^{4} - 97T_{23}^{3} + 996T_{23}^{2} - 2403T_{23} + 7921 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 15 T^{4} + \cdots + 400 \)
|
| $7$ |
\( T^{6} + 6 T^{4} + \cdots + 343 \)
|
| $11$ |
\( (T^{2} + T + 1)^{3} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 19321 \)
|
| $19$ |
\( T^{6} - 9 T^{5} + \cdots + 784 \)
|
| $23$ |
\( T^{6} - 3 T^{5} + \cdots + 7921 \)
|
| $29$ |
\( (T^{3} + 9 T^{2} + \cdots - 348)^{2} \)
|
| $31$ |
\( T^{6} - 6 T^{5} + \cdots + 36864 \)
|
| $37$ |
\( T^{6} + 21 T^{5} + \cdots + 51984 \)
|
| $41$ |
\( (T^{3} + 12 T^{2} + \cdots - 306)^{2} \)
|
| $43$ |
\( (T^{3} - 3 T^{2} + \cdots + 404)^{2} \)
|
| $47$ |
\( T^{6} - 3 T^{5} + \cdots + 81 \)
|
| $53$ |
\( (T^{2} - 8 T + 64)^{3} \)
|
| $59$ |
\( T^{6} + 9 T^{5} + \cdots + 2304 \)
|
| $61$ |
\( T^{6} - 6 T^{5} + \cdots + 9604 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 44944 \)
|
| $71$ |
\( (T^{3} + 9 T^{2} + \cdots - 108)^{2} \)
|
| $73$ |
\( T^{6} + 120 T^{4} + \cdots + 230400 \)
|
| $79$ |
\( T^{6} + 12 T^{5} + \cdots + 256 \)
|
| $83$ |
\( (T^{3} + 18 T^{2} + \cdots - 164)^{2} \)
|
| $89$ |
\( T^{6} + 12 T^{5} + \cdots + 256 \)
|
| $97$ |
\( (T - 7)^{6} \)
|
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