Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [663,2,Mod(16,663)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(663, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("663.16");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 663 = 3 \cdot 13 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 663.w (of order \(6\), 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 \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{24}^{6} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{24}^{7} + \zeta_{24} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_{2} \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( \beta_{3} \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).
| \(n\) | \(443\) | \(547\) | \(613\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 16.1 |
|
−1.20711 | − | 2.09077i | −0.866025 | + | 0.500000i | −1.91421 | + | 3.31552i | − | 1.00000i | 2.09077 | + | 1.20711i | −0.717439 | − | 0.414214i | 4.41421 | 0.500000 | − | 0.866025i | −2.09077 | + | 1.20711i | |||||||||||||||||||||||||||
| 16.2 | −1.20711 | − | 2.09077i | 0.866025 | − | 0.500000i | −1.91421 | + | 3.31552i | 1.00000i | −2.09077 | − | 1.20711i | 0.717439 | + | 0.414214i | 4.41421 | 0.500000 | − | 0.866025i | 2.09077 | − | 1.20711i | |||||||||||||||||||||||||||||
| 16.3 | 0.207107 | + | 0.358719i | −0.866025 | + | 0.500000i | 0.914214 | − | 1.58346i | − | 1.00000i | −0.358719 | − | 0.207107i | 4.18154 | + | 2.41421i | 1.58579 | 0.500000 | − | 0.866025i | 0.358719 | − | 0.207107i | ||||||||||||||||||||||||||||
| 16.4 | 0.207107 | + | 0.358719i | 0.866025 | − | 0.500000i | 0.914214 | − | 1.58346i | 1.00000i | 0.358719 | + | 0.207107i | −4.18154 | − | 2.41421i | 1.58579 | 0.500000 | − | 0.866025i | −0.358719 | + | 0.207107i | |||||||||||||||||||||||||||||
| 373.1 | −1.20711 | + | 2.09077i | −0.866025 | − | 0.500000i | −1.91421 | − | 3.31552i | 1.00000i | 2.09077 | − | 1.20711i | −0.717439 | + | 0.414214i | 4.41421 | 0.500000 | + | 0.866025i | −2.09077 | − | 1.20711i | |||||||||||||||||||||||||||||
| 373.2 | −1.20711 | + | 2.09077i | 0.866025 | + | 0.500000i | −1.91421 | − | 3.31552i | − | 1.00000i | −2.09077 | + | 1.20711i | 0.717439 | − | 0.414214i | 4.41421 | 0.500000 | + | 0.866025i | 2.09077 | + | 1.20711i | ||||||||||||||||||||||||||||
| 373.3 | 0.207107 | − | 0.358719i | −0.866025 | − | 0.500000i | 0.914214 | + | 1.58346i | 1.00000i | −0.358719 | + | 0.207107i | 4.18154 | − | 2.41421i | 1.58579 | 0.500000 | + | 0.866025i | 0.358719 | + | 0.207107i | |||||||||||||||||||||||||||||
| 373.4 | 0.207107 | − | 0.358719i | 0.866025 | + | 0.500000i | 0.914214 | + | 1.58346i | − | 1.00000i | 0.358719 | − | 0.207107i | −4.18154 | + | 2.41421i | 1.58579 | 0.500000 | + | 0.866025i | −0.358719 | − | 0.207107i | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 221.l | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 663.2.w.b | ✓ | 8 |
| 13.c | even | 3 | 1 | inner | 663.2.w.b | ✓ | 8 |
| 17.b | even | 2 | 1 | inner | 663.2.w.b | ✓ | 8 |
| 221.l | even | 6 | 1 | inner | 663.2.w.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.w.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 663.2.w.b | ✓ | 8 | 13.c | even | 3 | 1 | inner |
| 663.2.w.b | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 663.2.w.b | ✓ | 8 | 221.l | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( T^{8} - 24 T^{6} + \cdots + 256 \)
|
| $11$ |
\( T^{8} - 18 T^{6} + \cdots + 2401 \)
|
| $13$ |
\( (T^{2} + 2 T + 13)^{4} \)
|
| $17$ |
\( T^{8} - 2 T^{6} + \cdots + 83521 \)
|
| $19$ |
\( (T^{4} + 32 T^{2} + 1024)^{2} \)
|
| $23$ |
\( T^{8} - 66 T^{6} + \cdots + 83521 \)
|
| $29$ |
\( (T^{4} - 49 T^{2} + 2401)^{2} \)
|
| $31$ |
\( (T^{4} + 72 T^{2} + 784)^{2} \)
|
| $37$ |
\( T^{8} - 88 T^{6} + \cdots + 614656 \)
|
| $41$ |
\( (T^{4} - 36 T^{2} + 1296)^{2} \)
|
| $43$ |
\( (T^{4} + 6 T^{3} + 35 T^{2} + \cdots + 1)^{2} \)
|
| $47$ |
\( (T^{2} - 4 T - 124)^{4} \)
|
| $53$ |
\( (T^{2} - 8 T - 16)^{4} \)
|
| $59$ |
\( (T^{2} - 4 T + 16)^{4} \)
|
| $61$ |
\( T^{8} - 96 T^{6} + \cdots + 65536 \)
|
| $67$ |
\( (T^{4} - 14 T^{3} + \cdots + 1681)^{2} \)
|
| $71$ |
\( (T^{4} - 128 T^{2} + 16384)^{2} \)
|
| $73$ |
\( (T^{4} + 144 T^{2} + 3136)^{2} \)
|
| $79$ |
\( (T^{4} + 264 T^{2} + 4624)^{2} \)
|
| $83$ |
\( (T^{2} - 8)^{4} \)
|
| $89$ |
\( (T^{4} - 12 T^{3} + \cdots + 16)^{2} \)
|
| $97$ |
\( (T^{4} - 4 T^{2} + 16)^{2} \)
|
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