Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1098,2,Mod(379,1098)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1098, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1098.379");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1098 = 2 \cdot 3^{2} \cdot 61 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1098.f (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: | \(8.76757414194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.954288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 366) |
| 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
\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1098\mathbb{Z}\right)^\times\).
| \(n\) | \(245\) | \(307\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 379.1 |
|
−0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −1.16044 | + | 2.00994i | 0 | −0.0966262 | + | 0.167362i | 1.00000 | 0 | −1.16044 | − | 2.00994i | ||||||||||||||||||||||||||
| 379.2 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | −0.175970 | + | 0.304788i | 0 | 1.21903 | − | 2.11143i | 1.00000 | 0 | −0.175970 | − | 0.304788i | |||||||||||||||||||||||||||
| 379.3 | −0.500000 | + | 0.866025i | 0 | −0.500000 | − | 0.866025i | 1.83641 | − | 3.18076i | 0 | −2.12241 | + | 3.67612i | 1.00000 | 0 | 1.83641 | + | 3.18076i | |||||||||||||||||||||||||||
| 901.1 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −1.16044 | − | 2.00994i | 0 | −0.0966262 | − | 0.167362i | 1.00000 | 0 | −1.16044 | + | 2.00994i | |||||||||||||||||||||||||||
| 901.2 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | −0.175970 | − | 0.304788i | 0 | 1.21903 | + | 2.11143i | 1.00000 | 0 | −0.175970 | + | 0.304788i | |||||||||||||||||||||||||||
| 901.3 | −0.500000 | − | 0.866025i | 0 | −0.500000 | + | 0.866025i | 1.83641 | + | 3.18076i | 0 | −2.12241 | − | 3.67612i | 1.00000 | 0 | 1.83641 | − | 3.18076i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 61.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1098.2.f.e | 6 | |
| 3.b | odd | 2 | 1 | 366.2.e.e | ✓ | 6 | |
| 61.c | even | 3 | 1 | inner | 1098.2.f.e | 6 | |
| 183.k | odd | 6 | 1 | 366.2.e.e | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 366.2.e.e | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 366.2.e.e | ✓ | 6 | 183.k | odd | 6 | 1 | |
| 1098.2.f.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 1098.2.f.e | 6 | 61.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - T_{5}^{5} + 10T_{5}^{4} + 15T_{5}^{3} + 78T_{5}^{2} + 27T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(1098, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - T^{5} + 10 T^{4} + \cdots + 9 \)
|
| $7$ |
\( T^{6} + 2 T^{5} + \cdots + 4 \)
|
| $11$ |
\( (T^{3} + 6 T^{2} - 6 T - 54)^{2} \)
|
| $13$ |
\( T^{6} - 2 T^{5} + \cdots + 576 \)
|
| $17$ |
\( T^{6} + 24 T^{4} + \cdots + 1296 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots + 576 \)
|
| $23$ |
\( (T^{3} - 4 T^{2} + 6)^{2} \)
|
| $29$ |
\( (T^{2} - 3 T + 9)^{3} \)
|
| $31$ |
\( T^{6} - 2 T^{5} + \cdots + 5184 \)
|
| $37$ |
\( (T^{3} - T^{2} - 65 T - 57)^{2} \)
|
| $41$ |
\( (T^{3} + 3 T^{2} + \cdots - 603)^{2} \)
|
| $43$ |
\( T^{6} + 20 T^{5} + \cdots + 68644 \)
|
| $47$ |
\( T^{6} + 6 T^{5} + \cdots + 324 \)
|
| $53$ |
\( (T^{3} + 13 T^{2} + \cdots - 789)^{2} \)
|
| $59$ |
\( T^{6} + 96 T^{4} + \cdots + 82944 \)
|
| $61$ |
\( T^{6} - 5 T^{5} + \cdots + 226981 \)
|
| $67$ |
\( T^{6} + 2 T^{5} + \cdots + 252004 \)
|
| $71$ |
\( T^{6} - 2 T^{5} + \cdots + 576 \)
|
| $73$ |
\( T^{6} + T^{5} + \cdots + 514089 \)
|
| $79$ |
\( T^{6} + 14 T^{5} + \cdots + 64 \)
|
| $83$ |
\( T^{6} + 18 T^{5} + \cdots + 2125764 \)
|
| $89$ |
\( (T^{3} + 9 T^{2} + \cdots - 423)^{2} \)
|
| $97$ |
\( T^{6} - 15 T^{5} + \cdots + 17161 \)
|
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