Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [507,2,Mod(316,507)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(507, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("507.316");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 507 = 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 507.j (of order \(6\), degree \(2\), not 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: | \(4.04841538248\) |
| 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: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 39) |
| 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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{24}^{4} \)
|
| \(\beta_{2}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} + \zeta_{24} \)
|
| \(\beta_{4}\) | \(=\) |
\( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}^{2} \)
|
| \(\beta_{6}\) | \(=\) |
\( 2\zeta_{24}^{7} + 2\zeta_{24} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + \zeta_{24} \)
|
| \(\zeta_{24}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{3} ) / 4 \)
|
| \(\zeta_{24}^{2}\) | \(=\) |
\( ( \beta_{5} - \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{3}\) | \(=\) |
\( ( -\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} ) / 4 \)
|
| \(\zeta_{24}^{4}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{24}^{5}\) | \(=\) |
\( ( \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} ) / 4 \)
|
| \(\zeta_{24}^{6}\) | \(=\) |
\( ( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\zeta_{24}^{7}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{3} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).
| \(n\) | \(170\) | \(340\) |
| \(\chi(n)\) | \(1\) | \(1 - \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 316.1 |
|
−2.09077 | − | 1.20711i | −0.500000 | + | 0.866025i | 1.91421 | + | 3.31552i | − | 2.82843i | 2.09077 | − | 1.20711i | −2.44949 | + | 1.41421i | − | 4.41421i | −0.500000 | − | 0.866025i | −3.41421 | + | 5.91359i | ||||||||||||||||||||||||||
| 316.2 | −0.358719 | − | 0.207107i | −0.500000 | + | 0.866025i | −0.914214 | − | 1.58346i | − | 2.82843i | 0.358719 | − | 0.207107i | −2.44949 | + | 1.41421i | 1.58579i | −0.500000 | − | 0.866025i | −0.585786 | + | 1.01461i | ||||||||||||||||||||||||||||
| 316.3 | 0.358719 | + | 0.207107i | −0.500000 | + | 0.866025i | −0.914214 | − | 1.58346i | 2.82843i | −0.358719 | + | 0.207107i | 2.44949 | − | 1.41421i | − | 1.58579i | −0.500000 | − | 0.866025i | −0.585786 | + | 1.01461i | ||||||||||||||||||||||||||||
| 316.4 | 2.09077 | + | 1.20711i | −0.500000 | + | 0.866025i | 1.91421 | + | 3.31552i | 2.82843i | −2.09077 | + | 1.20711i | 2.44949 | − | 1.41421i | 4.41421i | −0.500000 | − | 0.866025i | −3.41421 | + | 5.91359i | |||||||||||||||||||||||||||||
| 361.1 | −2.09077 | + | 1.20711i | −0.500000 | − | 0.866025i | 1.91421 | − | 3.31552i | 2.82843i | 2.09077 | + | 1.20711i | −2.44949 | − | 1.41421i | 4.41421i | −0.500000 | + | 0.866025i | −3.41421 | − | 5.91359i | |||||||||||||||||||||||||||||
| 361.2 | −0.358719 | + | 0.207107i | −0.500000 | − | 0.866025i | −0.914214 | + | 1.58346i | 2.82843i | 0.358719 | + | 0.207107i | −2.44949 | − | 1.41421i | − | 1.58579i | −0.500000 | + | 0.866025i | −0.585786 | − | 1.01461i | ||||||||||||||||||||||||||||
| 361.3 | 0.358719 | − | 0.207107i | −0.500000 | − | 0.866025i | −0.914214 | + | 1.58346i | − | 2.82843i | −0.358719 | − | 0.207107i | 2.44949 | + | 1.41421i | 1.58579i | −0.500000 | + | 0.866025i | −0.585786 | − | 1.01461i | ||||||||||||||||||||||||||||
| 361.4 | 2.09077 | − | 1.20711i | −0.500000 | − | 0.866025i | 1.91421 | − | 3.31552i | − | 2.82843i | −2.09077 | − | 1.20711i | 2.44949 | + | 1.41421i | − | 4.41421i | −0.500000 | + | 0.866025i | −3.41421 | − | 5.91359i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):
|
\( T_{2}^{8} - 6T_{2}^{6} + 35T_{2}^{4} - 6T_{2}^{2} + 1 \)
|
|
\( T_{5}^{2} + 8 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 6 T^{6} + \cdots + 1 \)
|
| $3$ |
\( (T^{2} + T + 1)^{4} \)
|
| $5$ |
\( (T^{2} + 8)^{4} \)
|
| $7$ |
\( (T^{4} - 8 T^{2} + 64)^{2} \)
|
| $11$ |
\( (T^{4} - 4 T^{2} + 16)^{2} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( (T^{4} - 4 T^{3} + \cdots + 784)^{2} \)
|
| $19$ |
\( (T^{4} - 8 T^{2} + 64)^{2} \)
|
| $23$ |
\( (T^{2} + 4 T + 16)^{4} \)
|
| $29$ |
\( (T^{2} + 2 T + 4)^{4} \)
|
| $31$ |
\( (T^{4} + 48 T^{2} + 64)^{2} \)
|
| $37$ |
\( T^{8} - 72 T^{6} + \cdots + 614656 \)
|
| $41$ |
\( T^{8} - 144 T^{6} + \cdots + 9834496 \)
|
| $43$ |
\( (T^{4} - 8 T^{3} + \cdots + 256)^{2} \)
|
| $47$ |
\( (T^{4} + 136 T^{2} + 16)^{2} \)
|
| $53$ |
\( (T + 2)^{8} \)
|
| $59$ |
\( T^{8} - 72 T^{6} + \cdots + 614656 \)
|
| $61$ |
\( (T^{4} + 4 T^{3} + \cdots + 15376)^{2} \)
|
| $67$ |
\( T^{8} - 48 T^{6} + \cdots + 4096 \)
|
| $71$ |
\( (T^{4} - 4 T^{2} + 16)^{2} \)
|
| $73$ |
\( (T^{4} + 136 T^{2} + 16)^{2} \)
|
| $79$ |
\( (T^{2} - 128)^{4} \)
|
| $83$ |
\( (T^{4} + 72 T^{2} + 784)^{2} \)
|
| $89$ |
\( T^{8} - 304 T^{6} + \cdots + 342102016 \)
|
| $97$ |
\( T^{8} - 72 T^{6} + \cdots + 614656 \)
|
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