Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [798,2,Mod(457,798)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(798, 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("798.457");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 798.j (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: | \(6.37206208130\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.4406832.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 6\nu^{4} - 36\nu^{3} + 24\nu^{2} + 5\nu - 30 ) / 149 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -6\nu^{5} + 36\nu^{4} - 67\nu^{3} + 144\nu^{2} + 30\nu + 416 ) / 149 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 155 ) / 149 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 89\nu^{5} - 87\nu^{4} + 522\nu^{3} + 695\nu^{2} + 2088\nu + 435 ) / 149 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 6\beta_{2} - 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -6\beta_{5} + 19\beta_{4} + 6\beta_{3} - 12\beta _1 - 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -12\beta_{5} + 42\beta_{4} + 43\beta_{2} - 43\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(211\) | \(533\) |
| \(\chi(n)\) | \(-\beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 457.1 |
|
−0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −0.433099 | + | 0.750150i | 1.00000 | −1.69175 | − | 2.03420i | 1.00000 | −0.500000 | + | 0.866025i | −0.433099 | − | 0.750150i | ||||||||||||||||||||||
| 457.2 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 1.10538 | − | 1.91457i | 1.00000 | −1.16166 | + | 2.37709i | 1.00000 | −0.500000 | + | 0.866025i | 1.10538 | + | 1.91457i | |||||||||||||||||||||||
| 457.3 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 1.82772 | − | 3.16571i | 1.00000 | 2.35341 | − | 1.20891i | 1.00000 | −0.500000 | + | 0.866025i | 1.82772 | + | 3.16571i | |||||||||||||||||||||||
| 571.1 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −0.433099 | − | 0.750150i | 1.00000 | −1.69175 | + | 2.03420i | 1.00000 | −0.500000 | − | 0.866025i | −0.433099 | + | 0.750150i | |||||||||||||||||||||||
| 571.2 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 1.10538 | + | 1.91457i | 1.00000 | −1.16166 | − | 2.37709i | 1.00000 | −0.500000 | − | 0.866025i | 1.10538 | − | 1.91457i | |||||||||||||||||||||||
| 571.3 | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 1.82772 | + | 3.16571i | 1.00000 | 2.35341 | + | 1.20891i | 1.00000 | −0.500000 | − | 0.866025i | 1.82772 | − | 3.16571i | |||||||||||||||||||||||
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 | 798.2.j.i | ✓ | 6 |
| 7.c | even | 3 | 1 | inner | 798.2.j.i | ✓ | 6 |
| 7.c | even | 3 | 1 | 5586.2.a.bv | 3 | ||
| 7.d | odd | 6 | 1 | 5586.2.a.bu | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 798.2.j.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 798.2.j.i | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 5586.2.a.bu | 3 | 7.d | odd | 6 | 1 | ||
| 5586.2.a.bv | 3 | 7.c | even | 3 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):
|
\( T_{5}^{6} - 5T_{5}^{5} + 22T_{5}^{4} - 29T_{5}^{3} + 44T_{5}^{2} + 21T_{5} + 49 \)
|
|
\( T_{11}^{6} + T_{11}^{5} + 18T_{11}^{4} + 25T_{11}^{3} + 310T_{11}^{2} + 357T_{11} + 441 \)
|
|
\( T_{13}^{3} + 6T_{13}^{2} - 20T_{13} - 88 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( (T^{2} + T + 1)^{3} \)
|
| $5$ |
\( T^{6} - 5 T^{5} + \cdots + 49 \)
|
| $7$ |
\( T^{6} + T^{5} + \cdots + 343 \)
|
| $11$ |
\( T^{6} + T^{5} + \cdots + 441 \)
|
| $13$ |
\( (T^{3} + 6 T^{2} - 20 T - 88)^{2} \)
|
| $17$ |
\( T^{6} - 14 T^{5} + \cdots + 5476 \)
|
| $19$ |
\( (T^{2} - T + 1)^{3} \)
|
| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 91204 \)
|
| $29$ |
\( (T^{3} + T^{2} - 41 T + 43)^{2} \)
|
| $31$ |
\( T^{6} - 11 T^{5} + \cdots + 49 \)
|
| $37$ |
\( T^{6} - 4 T^{5} + \cdots + 37636 \)
|
| $41$ |
\( (T^{3} + 14 T^{2} + \cdots + 42)^{2} \)
|
| $43$ |
\( (T^{3} - 6 T^{2} + \cdots + 606)^{2} \)
|
| $47$ |
\( T^{6} + 4 T^{5} + \cdots + 20736 \)
|
| $53$ |
\( T^{6} - 9 T^{5} + \cdots + 10201 \)
|
| $59$ |
\( T^{6} - 7 T^{5} + \cdots + 3969 \)
|
| $61$ |
\( T^{6} + 10 T^{5} + \cdots + 3136 \)
|
| $67$ |
\( T^{6} - 12 T^{5} + \cdots + 1024 \)
|
| $71$ |
\( (T^{3} + 6 T^{2} + \cdots + 206)^{2} \)
|
| $73$ |
\( T^{6} - 2 T^{5} + \cdots + 38416 \)
|
| $79$ |
\( T^{6} - 15 T^{5} + \cdots + 1138489 \)
|
| $83$ |
\( (T^{3} + 19 T^{2} + \cdots - 3279)^{2} \)
|
| $89$ |
\( T^{6} - 14 T^{5} + \cdots + 1764 \)
|
| $97$ |
\( (T^{3} + 31 T^{2} + \cdots + 873)^{2} \)
|
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