Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [399,2,Mod(58,399)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(399, 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("399.58");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 399 = 3 \cdot 7 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 399.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: | \(3.18603104065\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.542936601.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 2x^{6} - 4x^{5} + x^{4} - 8x^{3} + 8x^{2} - 8x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + 2x^{6} - 4x^{5} + x^{4} - 8x^{3} + 8x^{2} - 8x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - \nu^{6} + 7\nu^{3} + 6\nu^{2} + 8\nu - 8 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} - 2\nu^{5} + 4\nu^{4} - \nu^{3} + 8\nu^{2} - 8\nu + 8 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{5} - \nu^{3} + 7\nu^{2} + 2\nu + 16 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 4\nu^{5} + \nu^{3} - 10\nu^{2} - 8\nu - 16 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3\nu^{7} + \nu^{6} + 6\nu^{5} + 3\nu^{3} - 20\nu^{2} - 4\nu - 32 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} + 2\nu^{5} - \nu^{4} + \nu^{3} - 5\nu^{2} + \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} - \beta_{6} + \beta_{3} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{4} + \beta_{2} - \beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} - 3\beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta _1 + 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 3\beta_{5} + 6\beta_{4} + 2\beta_{3} + 4\beta _1 - 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4\beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} + 4\beta_{3} - \beta_{2} - 3\beta _1 - 1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).
| \(n\) | \(115\) | \(134\) | \(211\) |
| \(\chi(n)\) | \(-1 + \beta_{4}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 58.1 |
|
−1.28863 | + | 2.23198i | −0.500000 | − | 0.866025i | −2.32116 | − | 4.02036i | −1.82116 | + | 3.15434i | 2.57727 | 0.967478 | + | 2.46252i | 6.80995 | −0.500000 | + | 0.866025i | −4.69361 | − | 8.12957i | ||||||||||||||||||||||||||||
| 58.2 | 0.266288 | − | 0.461225i | −0.500000 | − | 0.866025i | 0.858181 | + | 1.48641i | 1.35818 | − | 2.35244i | −0.532576 | 2.59189 | + | 0.531122i | 1.97925 | −0.500000 | + | 0.866025i | −0.723335 | − | 1.25285i | |||||||||||||||||||||||||||||
| 58.3 | 0.676830 | − | 1.17230i | −0.500000 | − | 0.866025i | 0.0838024 | + | 0.145150i | 0.583802 | − | 1.01118i | −1.35366 | 1.40697 | + | 2.24063i | 2.93420 | −0.500000 | + | 0.866025i | −0.790270 | − | 1.36879i | |||||||||||||||||||||||||||||
| 58.4 | 1.34552 | − | 2.33050i | −0.500000 | − | 0.866025i | −2.62083 | − | 4.53941i | −2.12083 | + | 3.67338i | −2.69103 | −1.96634 | − | 1.77017i | −8.72340 | −0.500000 | + | 0.866025i | 5.70721 | + | 9.88518i | |||||||||||||||||||||||||||||
| 172.1 | −1.28863 | − | 2.23198i | −0.500000 | + | 0.866025i | −2.32116 | + | 4.02036i | −1.82116 | − | 3.15434i | 2.57727 | 0.967478 | − | 2.46252i | 6.80995 | −0.500000 | − | 0.866025i | −4.69361 | + | 8.12957i | |||||||||||||||||||||||||||||
| 172.2 | 0.266288 | + | 0.461225i | −0.500000 | + | 0.866025i | 0.858181 | − | 1.48641i | 1.35818 | + | 2.35244i | −0.532576 | 2.59189 | − | 0.531122i | 1.97925 | −0.500000 | − | 0.866025i | −0.723335 | + | 1.25285i | |||||||||||||||||||||||||||||
| 172.3 | 0.676830 | + | 1.17230i | −0.500000 | + | 0.866025i | 0.0838024 | − | 0.145150i | 0.583802 | + | 1.01118i | −1.35366 | 1.40697 | − | 2.24063i | 2.93420 | −0.500000 | − | 0.866025i | −0.790270 | + | 1.36879i | |||||||||||||||||||||||||||||
| 172.4 | 1.34552 | + | 2.33050i | −0.500000 | + | 0.866025i | −2.62083 | + | 4.53941i | −2.12083 | − | 3.67338i | −2.69103 | −1.96634 | + | 1.77017i | −8.72340 | −0.500000 | − | 0.866025i | 5.70721 | − | 9.88518i | |||||||||||||||||||||||||||||
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 | 399.2.j.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1197.2.j.i | 8 | ||
| 7.c | even | 3 | 1 | inner | 399.2.j.e | ✓ | 8 |
| 7.c | even | 3 | 1 | 2793.2.a.bb | 4 | ||
| 7.d | odd | 6 | 1 | 2793.2.a.z | 4 | ||
| 21.g | even | 6 | 1 | 8379.2.a.ca | 4 | ||
| 21.h | odd | 6 | 1 | 1197.2.j.i | 8 | ||
| 21.h | odd | 6 | 1 | 8379.2.a.bz | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 399.2.j.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 399.2.j.e | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 1197.2.j.i | 8 | 3.b | odd | 2 | 1 | ||
| 1197.2.j.i | 8 | 21.h | odd | 6 | 1 | ||
| 2793.2.a.z | 4 | 7.d | odd | 6 | 1 | ||
| 2793.2.a.bb | 4 | 7.c | even | 3 | 1 | ||
| 8379.2.a.bz | 4 | 21.h | odd | 6 | 1 | ||
| 8379.2.a.ca | 4 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{7} + 10T_{2}^{6} - 14T_{2}^{5} + 67T_{2}^{4} - 98T_{2}^{3} + 139T_{2}^{2} - 65T_{2} + 25 \)
acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 25 \)
|
| $3$ |
\( (T^{2} + T + 1)^{4} \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 2401 \)
|
| $7$ |
\( T^{8} - 6 T^{7} + \cdots + 2401 \)
|
| $11$ |
\( T^{8} + 42 T^{6} + \cdots + 114921 \)
|
| $13$ |
\( (T^{4} - 42 T^{2} + \cdots + 339)^{2} \)
|
| $17$ |
\( (T^{2} + 3 T + 9)^{4} \)
|
| $19$ |
\( (T^{2} - T + 1)^{4} \)
|
| $23$ |
\( T^{8} - 5 T^{7} + \cdots + 1225 \)
|
| $29$ |
\( (T^{4} + 16 T^{3} + \cdots - 1379)^{2} \)
|
| $31$ |
\( T^{8} + 23 T^{7} + \cdots + 648025 \)
|
| $37$ |
\( T^{8} - T^{7} + \cdots + 49 \)
|
| $41$ |
\( (T^{4} + 5 T^{3} + \cdots + 619)^{2} \)
|
| $43$ |
\( (T^{4} - 5 T^{3} + \cdots + 211)^{2} \)
|
| $47$ |
\( T^{8} + 6 T^{7} + \cdots + 1071225 \)
|
| $53$ |
\( T^{8} - 12 T^{7} + \cdots + 9801 \)
|
| $59$ |
\( T^{8} - 15 T^{7} + \cdots + 690561 \)
|
| $61$ |
\( T^{8} + 7 T^{7} + \cdots + 49 \)
|
| $67$ |
\( T^{8} + 17 T^{7} + \cdots + 67092481 \)
|
| $71$ |
\( (T^{4} + 12 T^{3} + \cdots - 3045)^{2} \)
|
| $73$ |
\( T^{8} - 5 T^{7} + \cdots + 737881 \)
|
| $79$ |
\( T^{8} + 3 T^{7} + \cdots + 50168889 \)
|
| $83$ |
\( (T^{4} + 14 T^{3} + \cdots + 367)^{2} \)
|
| $89$ |
\( T^{8} + 7 T^{7} + \cdots + 2758921 \)
|
| $97$ |
\( (T^{4} - T^{3} - 36 T^{2} + \cdots + 43)^{2} \)
|
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