Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(50,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 7, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.50");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 663.bp (of order \(12\), degree \(4\), 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: | \(5.29408165401\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.1731891456.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{12}]$ |
$q$-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
\( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 1296 ) / 1040 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 181\nu ) / 260 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} + 116 ) / 65 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -29\nu^{6} + 325\nu^{4} - 1885\nu^{2} + 4176 ) / 1040 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9\nu^{7} - 65\nu^{5} + 585\nu^{3} - 256\nu ) / 1040 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9\nu^{7} - 65\nu^{5} + 377\nu^{3} - 256\nu ) / 832 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} + 5\beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -4\beta_{7} + 5\beta_{6} \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{5} + 29\beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -36\beta_{7} + 29\beta_{6} - 36\beta_{3} - 29\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 65\beta_{4} - 116 \)
|
| \(\nu^{7}\) | \(=\) |
\( -260\beta_{3} - 181\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).
| \(n\) | \(443\) | \(547\) | \(613\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 50.1 |
|
0 | 0.866025 | + | 1.50000i | −1.73205 | − | 1.00000i | −3.13312 | − | 3.13312i | 0 | 0 | 0 | −1.50000 | + | 2.59808i | 0 | ||||||||||||||||||||||||||||||||||
| 50.2 | 0 | 0.866025 | + | 1.50000i | −1.73205 | − | 1.00000i | 2.49915 | + | 2.49915i | 0 | 0 | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 254.1 | 0 | −0.866025 | + | 1.50000i | 1.73205 | − | 1.00000i | −1.93759 | − | 1.93759i | 0 | 0 | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 254.2 | 0 | −0.866025 | + | 1.50000i | 1.73205 | − | 1.00000i | −0.428432 | − | 0.428432i | 0 | 0 | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 305.1 | 0 | 0.866025 | − | 1.50000i | −1.73205 | + | 1.00000i | −3.13312 | + | 3.13312i | 0 | 0 | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 305.2 | 0 | 0.866025 | − | 1.50000i | −1.73205 | + | 1.00000i | 2.49915 | − | 2.49915i | 0 | 0 | 0 | −1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 509.1 | 0 | −0.866025 | − | 1.50000i | 1.73205 | + | 1.00000i | −1.93759 | + | 1.93759i | 0 | 0 | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
| 509.2 | 0 | −0.866025 | − | 1.50000i | 1.73205 | + | 1.00000i | −0.428432 | + | 0.428432i | 0 | 0 | 0 | −1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 51.c | odd | 2 | 1 | CM by \(\Q(\sqrt{-51}) \) |
| 13.f | odd | 12 | 1 | inner |
| 663.bp | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 663.2.bp.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 663.2.bp.b | yes | 8 | |
| 13.f | odd | 12 | 1 | inner | 663.2.bp.a | ✓ | 8 |
| 17.b | even | 2 | 1 | 663.2.bp.b | yes | 8 | |
| 39.k | even | 12 | 1 | 663.2.bp.b | yes | 8 | |
| 51.c | odd | 2 | 1 | CM | 663.2.bp.a | ✓ | 8 |
| 221.w | odd | 12 | 1 | 663.2.bp.b | yes | 8 | |
| 663.bp | even | 12 | 1 | inner | 663.2.bp.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.bp.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 663.2.bp.a | ✓ | 8 | 13.f | odd | 12 | 1 | inner |
| 663.2.bp.a | ✓ | 8 | 51.c | odd | 2 | 1 | CM |
| 663.2.bp.a | ✓ | 8 | 663.bp | even | 12 | 1 | inner |
| 663.2.bp.b | yes | 8 | 3.b | odd | 2 | 1 | |
| 663.2.bp.b | yes | 8 | 17.b | even | 2 | 1 | |
| 663.2.bp.b | yes | 8 | 39.k | even | 12 | 1 | |
| 663.2.bp.b | yes | 8 | 221.w | odd | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\):
|
\( T_{2} \)
|
|
\( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} + 6T_{5}^{5} + 173T_{5}^{4} + 948T_{5}^{3} + 2592T_{5}^{2} + 1872T_{5} + 676 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 3 T^{2} + 9)^{2} \)
|
| $5$ |
\( T^{8} + 6 T^{7} + \cdots + 676 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} - 18 T^{7} + \cdots + 114244 \)
|
| $13$ |
\( (T^{2} - T + 13)^{4} \)
|
| $17$ |
\( (T^{4} - 17 T^{2} + 289)^{2} \)
|
| $19$ |
\( T^{8} + 10 T^{7} + \cdots + 28561 \)
|
| $23$ |
\( (T^{4} + 15 T^{3} + \cdots + 36)^{2} \)
|
| $29$ |
\( (T^{4} - 12 T^{3} + \cdots + 361)^{2} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( T^{8} - 42 T^{7} + \cdots + 17850625 \)
|
| $43$ |
\( T^{8} - 137 T^{6} + \cdots + 4096 \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} + 16 T^{7} + \cdots + 73393489 \)
|
| $71$ |
\( T^{8} - 12 T^{7} + \cdots + 285610000 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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