from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(203, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,25]))
pari: [g,chi] = znchar(Mod(69,203))
Basic properties
| Modulus: | \(203\) | |
| Conductor: | \(203\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 203.r
\(\chi_{203}(27,\cdot)\) \(\chi_{203}(48,\cdot)\) \(\chi_{203}(55,\cdot)\) \(\chi_{203}(69,\cdot)\) \(\chi_{203}(76,\cdot)\) \(\chi_{203}(90,\cdot)\) \(\chi_{203}(97,\cdot)\) \(\chi_{203}(118,\cdot)\) \(\chi_{203}(153,\cdot)\) \(\chi_{203}(160,\cdot)\) \(\chi_{203}(188,\cdot)\) \(\chi_{203}(195,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.2070706293589565601613551437543564286910572644210741.1 |
Values on generators
\((59,176)\) → \((-1,e\left(\frac{25}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 203 }(69, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) |
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)