from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([19,0]))
pari: [g,chi] = znchar(Mod(56,405))
Basic properties
| Modulus: | \(405\) | |
| 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 \(\chi_{81}(56,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 405.u
\(\chi_{405}(11,\cdot)\) \(\chi_{405}(41,\cdot)\) \(\chi_{405}(56,\cdot)\) \(\chi_{405}(86,\cdot)\) \(\chi_{405}(101,\cdot)\) \(\chi_{405}(131,\cdot)\) \(\chi_{405}(146,\cdot)\) \(\chi_{405}(176,\cdot)\) \(\chi_{405}(191,\cdot)\) \(\chi_{405}(221,\cdot)\) \(\chi_{405}(236,\cdot)\) \(\chi_{405}(266,\cdot)\) \(\chi_{405}(281,\cdot)\) \(\chi_{405}(311,\cdot)\) \(\chi_{405}(326,\cdot)\) \(\chi_{405}(356,\cdot)\) \(\chi_{405}(371,\cdot)\) \(\chi_{405}(401,\cdot)\)
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
\((326,82)\) → \((e\left(\frac{19}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 405 }(56, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) |
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)