from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(527, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([25,36]))
pari: [g,chi] = znchar(Mod(178,527))
Basic properties
| Modulus: | \(527\) | |
| Conductor: | \(527\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 527.bc
\(\chi_{527}(15,\cdot)\) \(\chi_{527}(60,\cdot)\) \(\chi_{527}(77,\cdot)\) \(\chi_{527}(151,\cdot)\) \(\chi_{527}(178,\cdot)\) \(\chi_{527}(213,\cdot)\) \(\chi_{527}(240,\cdot)\) \(\chi_{527}(246,\cdot)\) \(\chi_{527}(263,\cdot)\) \(\chi_{527}(308,\cdot)\) \(\chi_{527}(325,\cdot)\) \(\chi_{527}(399,\cdot)\) \(\chi_{527}(457,\cdot)\) \(\chi_{527}(461,\cdot)\) \(\chi_{527}(519,\cdot)\) \(\chi_{527}(525,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.5685036093053617581480513152064675113048589266271429379917471760926790508296463148923889711866033.1 |
Values on generators
\((156,375)\) → \((e\left(\frac{5}{8}\right),e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 527 }(178, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
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)