Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [322,2,Mod(93,322)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("322.93");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 322 = 2 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 322.e (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: | \(2.57118294509\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.1767277521.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} - 2x^{7} + x^{6} - 10x^{5} + 38x^{4} - 40x^{3} + 64x^{2} - 38x + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 60\nu^{7} - 55\nu^{6} + 338\nu^{5} - 909\nu^{4} + 2308\nu^{3} - 5301\nu^{2} + 11263\nu - 6620 ) / 8102 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 74\nu^{7} - 743\nu^{6} + 957\nu^{5} - 716\nu^{4} + 8788\nu^{3} - 23147\nu^{2} + 13756\nu - 21668 ) / 8102 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -655\nu^{7} + 938\nu^{6} - 314\nu^{5} + 6885\nu^{4} - 22495\nu^{3} + 14321\nu^{2} - 38221\nu + 14204 ) / 8102 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -367\nu^{7} + 674\nu^{6} - 312\nu^{5} + 3332\nu^{4} - 13037\nu^{3} + 12372\nu^{2} - 18187\nu + 6734 ) / 4051 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -962\nu^{7} + 1557\nu^{6} - 288\nu^{5} + 9308\nu^{4} - 33224\nu^{3} + 25443\nu^{2} - 49196\nu + 18369 ) / 4051 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} + \cdots + 88979 ) / 8102 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} - \beta_{5} - 2\beta_{4} - 2\beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} - 3\beta_{6} - 4\beta_{4} + 2\beta_{3} + \beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{7} - 5\beta_{6} - 10\beta_{5} + 2\beta_{2} + 11\beta _1 - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( 15\beta_{6} - 22\beta_{5} - 20\beta_{4} - 2\beta_{3} - 4\beta_{2} - 2\beta _1 - 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 20\beta_{7} - 42\beta_{6} + \beta_{5} - 2\beta_{4} + 4\beta_{3} + 62\beta_{2} - 11\beta _1 + 34 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{7} + 5\beta_{6} - 115\beta_{5} + 88\beta_{4} - 62\beta_{3} + 68\beta_{2} + 30\beta _1 - 118 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).
| \(n\) | \(185\) | \(281\) |
| \(\chi(n)\) | \(-1 + \beta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 93.1 |
|
0.500000 | − | 0.866025i | −1.31242 | − | 2.27317i | −0.500000 | − | 0.866025i | −1.68584 | + | 2.91996i | −2.62484 | −1.55926 | + | 2.13746i | −1.00000 | −1.94488 | + | 3.36864i | 1.68584 | + | 2.91996i | ||||||||||||||||||||||||||||
| 93.2 | 0.500000 | − | 0.866025i | −0.937623 | − | 1.62401i | −0.500000 | − | 0.866025i | 0.603998 | − | 1.04616i | −1.87525 | 2.64562 | − | 0.0264594i | −1.00000 | −0.258274 | + | 0.447344i | −0.603998 | − | 1.04616i | |||||||||||||||||||||||||||||
| 93.3 | 0.500000 | − | 0.866025i | 0.319548 | + | 0.553474i | −0.500000 | − | 0.866025i | 0.268262 | − | 0.464643i | 0.639096 | 0.716975 | − | 2.54675i | −1.00000 | 1.29578 | − | 2.24435i | −0.268262 | − | 0.464643i | |||||||||||||||||||||||||||||
| 93.4 | 0.500000 | − | 0.866025i | 1.43049 | + | 2.47769i | −0.500000 | − | 0.866025i | −0.686423 | + | 1.18892i | 2.86099 | −2.30334 | + | 1.30178i | −1.00000 | −2.59262 | + | 4.49055i | 0.686423 | + | 1.18892i | |||||||||||||||||||||||||||||
| 277.1 | 0.500000 | + | 0.866025i | −1.31242 | + | 2.27317i | −0.500000 | + | 0.866025i | −1.68584 | − | 2.91996i | −2.62484 | −1.55926 | − | 2.13746i | −1.00000 | −1.94488 | − | 3.36864i | 1.68584 | − | 2.91996i | |||||||||||||||||||||||||||||
| 277.2 | 0.500000 | + | 0.866025i | −0.937623 | + | 1.62401i | −0.500000 | + | 0.866025i | 0.603998 | + | 1.04616i | −1.87525 | 2.64562 | + | 0.0264594i | −1.00000 | −0.258274 | − | 0.447344i | −0.603998 | + | 1.04616i | |||||||||||||||||||||||||||||
| 277.3 | 0.500000 | + | 0.866025i | 0.319548 | − | 0.553474i | −0.500000 | + | 0.866025i | 0.268262 | + | 0.464643i | 0.639096 | 0.716975 | + | 2.54675i | −1.00000 | 1.29578 | + | 2.24435i | −0.268262 | + | 0.464643i | |||||||||||||||||||||||||||||
| 277.4 | 0.500000 | + | 0.866025i | 1.43049 | − | 2.47769i | −0.500000 | + | 0.866025i | −0.686423 | − | 1.18892i | 2.86099 | −2.30334 | − | 1.30178i | −1.00000 | −2.59262 | − | 4.49055i | 0.686423 | − | 1.18892i | |||||||||||||||||||||||||||||
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 | 322.2.e.c | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 322.2.e.c | ✓ | 8 |
| 7.c | even | 3 | 1 | 2254.2.a.v | 4 | ||
| 7.d | odd | 6 | 1 | 2254.2.a.q | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 322.2.e.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 322.2.e.c | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 2254.2.a.q | 4 | 7.d | odd | 6 | 1 | ||
| 2254.2.a.v | 4 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + T_{3}^{7} + 10T_{3}^{6} + 9T_{3}^{5} + 81T_{3}^{4} + 63T_{3}^{3} + 162T_{3}^{2} - 81T_{3} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + 3 T^{7} + \cdots + 9 \)
|
| $7$ |
\( T^{8} + T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots + 3969 \)
|
| $13$ |
\( (T^{4} + T^{3} - 24 T^{2} + \cdots - 49)^{2} \)
|
| $17$ |
\( T^{8} + 15 T^{7} + \cdots + 15129 \)
|
| $19$ |
\( T^{8} - T^{7} + \cdots + 3481 \)
|
| $23$ |
\( (T^{2} + T + 1)^{4} \)
|
| $29$ |
\( (T^{4} - 6 T^{3} - 15 T^{2} + \cdots - 81)^{2} \)
|
| $31$ |
\( T^{8} + 8 T^{7} + \cdots + 7569 \)
|
| $37$ |
\( T^{8} + 8 T^{7} + \cdots + 14432401 \)
|
| $41$ |
\( (T^{4} - 9 T^{3} + \cdots - 567)^{2} \)
|
| $43$ |
\( (T^{4} - 14 T^{3} + \cdots + 767)^{2} \)
|
| $47$ |
\( T^{8} + 9 T^{7} + \cdots + 408321 \)
|
| $53$ |
\( T^{8} - 3 T^{7} + \cdots + 11350161 \)
|
| $59$ |
\( T^{8} + 12 T^{7} + \cdots + 6561 \)
|
| $61$ |
\( T^{8} + 11 T^{7} + \cdots + 58537801 \)
|
| $67$ |
\( T^{8} - T^{7} + \cdots + 790321 \)
|
| $71$ |
\( (T^{4} + 3 T^{3} + \cdots - 369)^{2} \)
|
| $73$ |
\( T^{8} - 4 T^{7} + \cdots + 2152089 \)
|
| $79$ |
\( T^{8} + \cdots + 1315730529 \)
|
| $83$ |
\( (T^{4} - 12 T^{3} + \cdots - 1029)^{2} \)
|
| $89$ |
\( T^{8} + 27 T^{7} + \cdots + 531441 \)
|
| $97$ |
\( (T^{4} - 2 T^{3} + \cdots + 2427)^{2} \)
|
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