from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,31]))
pari: [g,chi] = znchar(Mod(59,740))
Basic properties
| Modulus: | \(740\) | |
| Conductor: | \(740\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 740.ca
\(\chi_{740}(19,\cdot)\) \(\chi_{740}(39,\cdot)\) \(\chi_{740}(59,\cdot)\) \(\chi_{740}(79,\cdot)\) \(\chi_{740}(239,\cdot)\) \(\chi_{740}(279,\cdot)\) \(\chi_{740}(439,\cdot)\) \(\chi_{740}(459,\cdot)\) \(\chi_{740}(479,\cdot)\) \(\chi_{740}(499,\cdot)\) \(\chi_{740}(579,\cdot)\) \(\chi_{740}(679,\cdot)\)
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: | 36.36.2021157996966699236888348986167488408933250634681517576814592000000000000000000.1 |
Values on generators
\((371,297,261)\) → \((-1,-1,e\left(\frac{31}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 740 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)