from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(567, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([37,0]))
pari: [g,chi] = znchar(Mod(29,567))
Basic properties
| Modulus: | \(567\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 567.bp
\(\chi_{567}(29,\cdot)\) \(\chi_{567}(50,\cdot)\) \(\chi_{567}(92,\cdot)\) \(\chi_{567}(113,\cdot)\) \(\chi_{567}(155,\cdot)\) \(\chi_{567}(176,\cdot)\) \(\chi_{567}(218,\cdot)\) \(\chi_{567}(239,\cdot)\) \(\chi_{567}(281,\cdot)\) \(\chi_{567}(302,\cdot)\) \(\chi_{567}(344,\cdot)\) \(\chi_{567}(365,\cdot)\) \(\chi_{567}(407,\cdot)\) \(\chi_{567}(428,\cdot)\) \(\chi_{567}(470,\cdot)\) \(\chi_{567}(491,\cdot)\) \(\chi_{567}(533,\cdot)\) \(\chi_{567}(554,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((407,325)\) → \((e\left(\frac{37}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 567 }(29, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)