from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,9,26]))
pari: [g,chi] = znchar(Mod(307,760))
Basic properties
| Modulus: | \(760\) | |
| Conductor: | \(760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 760.cn
\(\chi_{760}(3,\cdot)\) \(\chi_{760}(67,\cdot)\) \(\chi_{760}(147,\cdot)\) \(\chi_{760}(203,\cdot)\) \(\chi_{760}(243,\cdot)\) \(\chi_{760}(307,\cdot)\) \(\chi_{760}(363,\cdot)\) \(\chi_{760}(507,\cdot)\) \(\chi_{760}(523,\cdot)\) \(\chi_{760}(547,\cdot)\) \(\chi_{760}(603,\cdot)\) \(\chi_{760}(667,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1 |
Values on generators
\((191,381,457,401)\) → \((-1,-1,i,e\left(\frac{13}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 760 }(307, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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)