Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(203,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.203");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 663.q (of order \(4\), 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: | \(5.29408165401\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.12745506816.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 49x^{4} + 2401 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} - 49x^{4} + 2401 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 49\nu ) / 49 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} ) / 343 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2\nu^{4} - 49 ) / 49 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 49\nu ) / 49 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 98\nu^{2} ) / 343 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2\nu^{7} - 49\nu^{3} ) / 343 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{6} + 7\beta_{3} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 49\beta_{4} + 49 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -49\beta_{5} + 49\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( 343\beta_{3} \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 343\beta_{7} + 343\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 203.1 |
|
−1.87083 | + | 1.87083i | −1.73205 | − | 5.00000i | −1.73205 | − | 1.73205i | 3.24037 | − | 3.24037i | 0 | 5.61249 | + | 5.61249i | 3.00000 | 6.48074 | |||||||||||||||||||||||||||||||||
| 203.2 | −1.87083 | + | 1.87083i | 1.73205 | − | 5.00000i | 1.73205 | + | 1.73205i | −3.24037 | + | 3.24037i | 0 | 5.61249 | + | 5.61249i | 3.00000 | −6.48074 | ||||||||||||||||||||||||||||||||||
| 203.3 | 1.87083 | − | 1.87083i | −1.73205 | − | 5.00000i | −1.73205 | − | 1.73205i | −3.24037 | + | 3.24037i | 0 | −5.61249 | − | 5.61249i | 3.00000 | −6.48074 | ||||||||||||||||||||||||||||||||||
| 203.4 | 1.87083 | − | 1.87083i | 1.73205 | − | 5.00000i | 1.73205 | + | 1.73205i | 3.24037 | − | 3.24037i | 0 | −5.61249 | − | 5.61249i | 3.00000 | 6.48074 | ||||||||||||||||||||||||||||||||||
| 356.1 | −1.87083 | − | 1.87083i | −1.73205 | 5.00000i | −1.73205 | + | 1.73205i | 3.24037 | + | 3.24037i | 0 | 5.61249 | − | 5.61249i | 3.00000 | 6.48074 | |||||||||||||||||||||||||||||||||||
| 356.2 | −1.87083 | − | 1.87083i | 1.73205 | 5.00000i | 1.73205 | − | 1.73205i | −3.24037 | − | 3.24037i | 0 | 5.61249 | − | 5.61249i | 3.00000 | −6.48074 | |||||||||||||||||||||||||||||||||||
| 356.3 | 1.87083 | + | 1.87083i | −1.73205 | 5.00000i | −1.73205 | + | 1.73205i | −3.24037 | − | 3.24037i | 0 | −5.61249 | + | 5.61249i | 3.00000 | −6.48074 | |||||||||||||||||||||||||||||||||||
| 356.4 | 1.87083 | + | 1.87083i | 1.73205 | 5.00000i | 1.73205 | − | 1.73205i | 3.24037 | + | 3.24037i | 0 | −5.61249 | + | 5.61249i | 3.00000 | 6.48074 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 39.f | even | 4 | 1 | inner |
| 51.c | odd | 2 | 1 | inner |
| 221.g | odd | 4 | 1 | inner |
| 663.q | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 663.2.q.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 663.2.q.a | ✓ | 8 |
| 13.d | odd | 4 | 1 | inner | 663.2.q.a | ✓ | 8 |
| 17.b | even | 2 | 1 | inner | 663.2.q.a | ✓ | 8 |
| 39.f | even | 4 | 1 | inner | 663.2.q.a | ✓ | 8 |
| 51.c | odd | 2 | 1 | inner | 663.2.q.a | ✓ | 8 |
| 221.g | odd | 4 | 1 | inner | 663.2.q.a | ✓ | 8 |
| 663.q | even | 4 | 1 | inner | 663.2.q.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.q.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 663.2.q.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 663.2.q.a | ✓ | 8 | 13.d | odd | 4 | 1 | inner |
| 663.2.q.a | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 663.2.q.a | ✓ | 8 | 39.f | even | 4 | 1 | inner |
| 663.2.q.a | ✓ | 8 | 51.c | odd | 2 | 1 | inner |
| 663.2.q.a | ✓ | 8 | 221.g | odd | 4 | 1 | inner |
| 663.2.q.a | ✓ | 8 | 663.q | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 49 \)
acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 49)^{2} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( (T^{4} + 36)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 36)^{2} \)
|
| $13$ |
\( (T^{2} + 6 T + 13)^{4} \)
|
| $17$ |
\( (T^{4} - 22 T^{2} + 289)^{2} \)
|
| $19$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $23$ |
\( (T^{2} + 12)^{4} \)
|
| $29$ |
\( (T^{2} - 12)^{4} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( (T^{4} + 7056)^{2} \)
|
| $41$ |
\( (T^{4} + 2916)^{2} \)
|
| $43$ |
\( (T^{2} + 16)^{4} \)
|
| $47$ |
\( (T^{4} + 784)^{2} \)
|
| $53$ |
\( (T^{2} + 56)^{4} \)
|
| $59$ |
\( (T^{4} + 784)^{2} \)
|
| $61$ |
\( (T^{2} + 168)^{4} \)
|
| $67$ |
\( (T^{2} + 14 T + 98)^{4} \)
|
| $71$ |
\( (T^{4} + 2916)^{2} \)
|
| $73$ |
\( (T^{4} + 7056)^{2} \)
|
| $79$ |
\( (T^{2} + 168)^{4} \)
|
| $83$ |
\( (T^{4} + 784)^{2} \)
|
| $89$ |
\( (T^{4} + 12544)^{2} \)
|
| $97$ |
\( (T^{4} + 7056)^{2} \)
|
| show more | |
| show less |