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: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.856615824.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| 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
\( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 9\nu^{4} + 2\nu^{3} + 22\nu^{2} + 10\nu + 12 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 11\nu^{5} - 34\nu^{3} - 2\nu^{2} - 20\nu - 4 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 10\nu^{5} + 2\nu^{4} - 29\nu^{3} + 10\nu^{2} - 26\nu + 2 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 11\nu^{5} - 34\nu^{3} + 2\nu^{2} - 20\nu + 4 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 29\nu^{3} - 20\nu^{2} + 20\nu - 6 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 11\nu^{5} - 4\nu^{4} + 34\nu^{3} - 22\nu^{2} + 20\nu - 12 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} - \beta_{6} - 2\beta_{4} - \beta_{2} + \beta _1 - 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{3} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} + 8\beta_{6} + 10\beta_{4} + 3\beta_{3} + 8\beta_{2} - 2\beta _1 + 2 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} - 6\beta_{5} + 5\beta_{3} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -25\beta_{7} - 46\beta_{6} - 50\beta_{4} - 21\beta_{3} - 46\beta_{2} + 16\beta _1 - 10 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{7} - 2\beta_{6} + 32\beta_{5} - 25\beta_{3} + 2\beta_{2} - 2\beta _1 - 38 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 125\beta_{7} + 254\beta_{6} - 6\beta_{5} + 250\beta_{4} + 123\beta_{3} + 254\beta_{2} - 128\beta _1 + 62 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(211\) | \(533\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 457.1 |
|
0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −1.24624 | + | 2.15855i | 1.00000 | 2.47720 | + | 0.929227i | −1.00000 | −0.500000 | + | 0.866025i | 1.24624 | + | 2.15855i | ||||||||||||||||||||||||||||
| 457.2 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −1.05903 | + | 1.83430i | 1.00000 | −1.11699 | − | 2.39840i | −1.00000 | −0.500000 | + | 0.866025i | 1.05903 | + | 1.83430i | |||||||||||||||||||||||||||||
| 457.3 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 0.574618 | − | 0.995268i | 1.00000 | 0.00953166 | + | 2.64573i | −1.00000 | −0.500000 | + | 0.866025i | −0.574618 | − | 0.995268i | |||||||||||||||||||||||||||||
| 457.4 | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | 1.73065 | − | 2.99758i | 1.00000 | −2.36975 | − | 1.17656i | −1.00000 | −0.500000 | + | 0.866025i | −1.73065 | − | 2.99758i | |||||||||||||||||||||||||||||
| 571.1 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −1.24624 | − | 2.15855i | 1.00000 | 2.47720 | − | 0.929227i | −1.00000 | −0.500000 | − | 0.866025i | 1.24624 | − | 2.15855i | |||||||||||||||||||||||||||||
| 571.2 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −1.05903 | − | 1.83430i | 1.00000 | −1.11699 | + | 2.39840i | −1.00000 | −0.500000 | − | 0.866025i | 1.05903 | − | 1.83430i | |||||||||||||||||||||||||||||
| 571.3 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 0.574618 | + | 0.995268i | 1.00000 | 0.00953166 | − | 2.64573i | −1.00000 | −0.500000 | − | 0.866025i | −0.574618 | + | 0.995268i | |||||||||||||||||||||||||||||
| 571.4 | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | 1.73065 | + | 2.99758i | 1.00000 | −2.36975 | + | 1.17656i | −1.00000 | −0.500000 | − | 0.866025i | −1.73065 | + | 2.99758i | |||||||||||||||||||||||||||||
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.l | ✓ | 8 |
| 7.c | even | 3 | 1 | inner | 798.2.j.l | ✓ | 8 |
| 7.c | even | 3 | 1 | 5586.2.a.bw | 4 | ||
| 7.d | odd | 6 | 1 | 5586.2.a.bz | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 798.2.j.l | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 798.2.j.l | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 5586.2.a.bw | 4 | 7.c | even | 3 | 1 | ||
| 5586.2.a.bz | 4 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\):
|
\( T_{5}^{8} + 12T_{5}^{6} + 12T_{5}^{5} + 123T_{5}^{4} + 72T_{5}^{3} + 288T_{5}^{2} - 126T_{5} + 441 \)
|
|
\( T_{11}^{8} - 2T_{11}^{7} + 16T_{11}^{6} - 20T_{11}^{5} + 181T_{11}^{4} - 236T_{11}^{3} + 568T_{11}^{2} + 154T_{11} + 49 \)
|
|
\( T_{13}^{4} - 48T_{13}^{2} - 48T_{13} + 336 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( (T^{2} - T + 1)^{4} \)
|
| $5$ |
\( T^{8} + 12 T^{6} + \cdots + 441 \)
|
| $7$ |
\( T^{8} + 2 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} - 2 T^{7} + \cdots + 49 \)
|
| $13$ |
\( (T^{4} - 48 T^{2} + \cdots + 336)^{2} \)
|
| $17$ |
\( T^{8} + 10 T^{7} + \cdots + 204304 \)
|
| $19$ |
\( (T^{2} + T + 1)^{4} \)
|
| $23$ |
\( T^{8} - 5 T^{7} + \cdots + 850084 \)
|
| $29$ |
\( (T^{4} + 3 T^{3} + \cdots + 1296)^{2} \)
|
| $31$ |
\( T^{8} + 9 T^{7} + \cdots + 1140624 \)
|
| $37$ |
\( T^{8} - 14 T^{7} + \cdots + 64 \)
|
| $41$ |
\( (T^{4} - 4 T^{3} + \cdots - 344)^{2} \)
|
| $43$ |
\( (T^{4} - 21 T^{3} + \cdots - 654)^{2} \)
|
| $47$ |
\( T^{8} + 7 T^{7} + \cdots + 65536 \)
|
| $53$ |
\( T^{8} - 7 T^{7} + \cdots + 952576 \)
|
| $59$ |
\( T^{8} + 7 T^{7} + \cdots + 84676804 \)
|
| $61$ |
\( T^{8} + 23 T^{7} + \cdots + 506944 \)
|
| $67$ |
\( T^{8} + 6 T^{7} + \cdots + 63489024 \)
|
| $71$ |
\( (T^{4} - 2 T^{3} - 126 T^{2} + \cdots + 28)^{2} \)
|
| $73$ |
\( T^{8} - 5 T^{7} + \cdots + 2808976 \)
|
| $79$ |
\( T^{8} - 11 T^{7} + \cdots + 29584 \)
|
| $83$ |
\( (T^{4} - 14 T^{3} + \cdots + 43)^{2} \)
|
| $89$ |
\( T^{8} + 10 T^{7} + \cdots + 23464336 \)
|
| $97$ |
\( (T^{4} + T^{3} - 69 T^{2} + \cdots + 1048)^{2} \)
|
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