Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1584,3,Mod(881,1584)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1584, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1584.881");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1584.i (of order \(2\), degree \(1\), not 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: | \(43.1608738747\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.65306824704.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 99) |
| 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
\( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + x^{4} - 40x^{3} + 36x^{2} - 12x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 17\nu^{3} - 20\nu^{2} + 29\nu - 3 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{7} + 11\nu^{6} + 60\nu^{5} - 83\nu^{4} - 170\nu^{3} - 21\nu^{2} + 21\nu + 54 ) / 324 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{7} + 11\nu^{6} + 60\nu^{5} - 83\nu^{4} - 170\nu^{3} + 303\nu^{2} + 21\nu - 756 ) / 162 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 19\nu^{7} - 53\nu^{6} - 186\nu^{5} + 503\nu^{4} + 716\nu^{3} - 1563\nu^{2} - 219\nu + 1080 ) / 324 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{7} - 28\nu^{6} - 30\nu^{5} + 145\nu^{4} + 40\nu^{3} - 219\nu^{2} + 300\nu - 108 ) / 81 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -35\nu^{7} + 136\nu^{6} + 57\nu^{5} - 577\nu^{4} - 121\nu^{3} + 1191\nu^{2} - 1596\nu + 513 ) / 324 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 47\nu^{7} - 178\nu^{6} - 183\nu^{5} + 997\nu^{4} + 667\nu^{3} - 2127\nu^{2} - 546\nu - 27 ) / 324 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + 3 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 2\beta_{2} + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} - 7\beta_{6} - 9\beta_{5} + 4\beta_{4} + 7\beta_{3} - 16\beta_{2} + 21 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{6} - 6\beta_{5} + 2\beta_{4} + 5\beta_{3} - 22\beta_{2} + 2\beta _1 + 17 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{7} - 49\beta_{6} - 90\beta_{5} + 7\beta_{4} + 25\beta_{3} - 184\beta_{2} + 15\beta _1 + 93 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -2\beta_{7} - 31\beta_{6} - 75\beta_{5} + 6\beta_{4} + 5\beta_{3} - 195\beta_{2} + 19\beta _1 - 1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 29\beta_{7} - 328\beta_{6} - 693\beta_{5} - 77\beta_{4} - 116\beta_{3} - 1588\beta_{2} + 147\beta _1 - 312 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(353\) | \(991\) | \(1189\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 881.1 |
|
0 | 0 | 0 | − | 7.62670i | 0 | 2.45638 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 881.2 | 0 | 0 | 0 | − | 6.21249i | 0 | −11.1468 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 881.3 | 0 | 0 | 0 | − | 6.10332i | 0 | −2.61086 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 881.4 | 0 | 0 | 0 | − | 4.68911i | 0 | 3.30128 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 881.5 | 0 | 0 | 0 | 4.68911i | 0 | 3.30128 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 881.6 | 0 | 0 | 0 | 6.10332i | 0 | −2.61086 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 881.7 | 0 | 0 | 0 | 6.21249i | 0 | −11.1468 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 881.8 | 0 | 0 | 0 | 7.62670i | 0 | 2.45638 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1584.3.i.b | 8 | |
| 3.b | odd | 2 | 1 | inner | 1584.3.i.b | 8 | |
| 4.b | odd | 2 | 1 | 99.3.b.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 99.3.b.a | ✓ | 8 | |
| 44.c | even | 2 | 1 | 1089.3.b.g | 8 | ||
| 132.d | odd | 2 | 1 | 1089.3.b.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 99.3.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 99.3.b.a | ✓ | 8 | 12.b | even | 2 | 1 | |
| 1089.3.b.g | 8 | 44.c | even | 2 | 1 | ||
| 1089.3.b.g | 8 | 132.d | odd | 2 | 1 | ||
| 1584.3.i.b | 8 | 1.a | even | 1 | 1 | trivial | |
| 1584.3.i.b | 8 | 3.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 156T_{5}^{6} + 8796T_{5}^{4} + 212240T_{5}^{2} + 1838736 \)
acting on \(S_{3}^{\mathrm{new}}(1584, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 156 T^{6} + \cdots + 1838736 \)
|
| $7$ |
\( (T^{4} + 8 T^{3} + \cdots + 236)^{2} \)
|
| $11$ |
\( (T^{2} + 11)^{4} \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 2828)^{2} \)
|
| $17$ |
\( T^{8} + 1080 T^{6} + \cdots + 472801536 \)
|
| $19$ |
\( (T^{4} + 20 T^{3} + \cdots - 37908)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 11052737424 \)
|
| $29$ |
\( T^{8} + \cdots + 125180100864 \)
|
| $31$ |
\( (T^{4} - 28 T^{3} + \cdots + 244016)^{2} \)
|
| $37$ |
\( (T^{4} - 68 T^{3} + \cdots - 914256)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 85142571264 \)
|
| $43$ |
\( (T^{4} - 52 T^{3} + \cdots - 3241332)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 12929518743696 \)
|
| $53$ |
\( T^{8} + \cdots + 26983975824 \)
|
| $59$ |
\( T^{8} + \cdots + 1028100096 \)
|
| $61$ |
\( (T^{4} + 4 T^{3} + \cdots + 1712844)^{2} \)
|
| $67$ |
\( (T^{4} + 56 T^{3} + \cdots - 2846576)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 13312019262096 \)
|
| $73$ |
\( (T^{4} - 224 T^{3} + \cdots - 37468144)^{2} \)
|
| $79$ |
\( (T^{4} + 224 T^{3} + \cdots - 4606964)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 177188471398656 \)
|
| $89$ |
\( T^{8} + \cdots + 13\!\cdots\!84 \)
|
| $97$ |
\( (T^{4} + 76 T^{3} + \cdots - 9076976)^{2} \)
|
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