Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [798,2,Mod(463,798)]
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, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("798.463");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 798.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: | \(6.37206208130\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.70858800.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 18x^{4} + 81x^{2} + 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} + 18x^{4} + 81x^{2} + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 9\nu + 4 ) / 8 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 13\nu^{2} + 4\nu + 24 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} - 13\nu^{2} + 4\nu - 24 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 15\nu^{3} + 4\nu^{2} + 38\nu + 24 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{4} - 9\beta_{3} + 16\beta _1 - 8 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{4} + 2\beta_{3} - 13\beta_{2} + 54 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 16\beta_{5} + 97\beta_{4} + 97\beta_{3} - 8\beta_{2} - 240\beta _1 + 120 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(211\) | \(533\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 463.1 |
|
−0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.00000 | 1.00000 | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | ||||||||||||||||||||||
| 463.2 | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.00000 | 1.00000 | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||
| 463.3 | −0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | −1.00000 | 1.00000 | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | |||||||||||||||||||||||
| 505.1 | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | −1.00000 | 1.00000 | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||
| 505.2 | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | −1.00000 | 1.00000 | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||
| 505.3 | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | −1.00000 | 1.00000 | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.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.k.n | ✓ | 6 |
| 3.b | odd | 2 | 1 | 2394.2.o.t | 6 | ||
| 19.c | even | 3 | 1 | inner | 798.2.k.n | ✓ | 6 |
| 57.h | odd | 6 | 1 | 2394.2.o.t | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 798.2.k.n | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 798.2.k.n | ✓ | 6 | 19.c | even | 3 | 1 | inner |
| 2394.2.o.t | 6 | 3.b | odd | 2 | 1 | ||
| 2394.2.o.t | 6 | 57.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):
|
\( T_{5}^{2} + T_{5} + 1 \)
|
|
\( T_{11}^{3} + 3T_{11}^{2} - 24T_{11} - 32 \)
|
|
\( T_{13}^{6} + 27T_{13}^{4} - 12T_{13}^{3} + 729T_{13}^{2} - 162T_{13} + 36 \)
|
|
\( T_{17}^{6} - 3T_{17}^{5} + 69T_{17}^{4} - 220T_{17}^{3} + 4200T_{17}^{2} - 12000T_{17} + 40000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{3} \)
|
| $3$ |
\( (T^{2} - T + 1)^{3} \)
|
| $5$ |
\( (T^{2} + T + 1)^{3} \)
|
| $7$ |
\( (T + 1)^{6} \)
|
| $11$ |
\( (T^{3} + 3 T^{2} - 24 T - 32)^{2} \)
|
| $13$ |
\( T^{6} + 27 T^{4} + \cdots + 36 \)
|
| $17$ |
\( T^{6} - 3 T^{5} + \cdots + 40000 \)
|
| $19$ |
\( T^{6} + 3 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} + 6 T^{5} + \cdots + 65536 \)
|
| $29$ |
\( (T^{2} + 4 T + 16)^{3} \)
|
| $31$ |
\( (T^{3} - 108 T - 48)^{2} \)
|
| $37$ |
\( (T^{3} + 3 T^{2} - 24 T - 32)^{2} \)
|
| $41$ |
\( T^{6} - 9 T^{5} + \cdots + 2304 \)
|
| $43$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $47$ |
\( T^{6} + 72 T^{4} + \cdots + 9216 \)
|
| $53$ |
\( T^{6} + 108 T^{4} + \cdots + 82944 \)
|
| $59$ |
\( T^{6} + 12 T^{5} + \cdots + 857476 \)
|
| $61$ |
\( T^{6} + 18 T^{5} + \cdots + 3600 \)
|
| $67$ |
\( T^{6} + 12 T^{5} + \cdots + 430336 \)
|
| $71$ |
\( T^{6} + 6 T^{5} + \cdots + 506944 \)
|
| $73$ |
\( T^{6} - 6 T^{5} + \cdots + 6400 \)
|
| $79$ |
\( T^{6} \)
|
| $83$ |
\( (T^{3} - 6 T^{2} - 51 T - 20)^{2} \)
|
| $89$ |
\( T^{6} - 9 T^{5} + \cdots + 90000 \)
|
| $97$ |
\( T^{6} + 12 T^{5} + \cdots + 102400 \)
|
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