from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,1]))
pari: [g,chi] = znchar(Mod(39,740))
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
\((371,297,261)\) → \((-1,-1,e\left(\frac{1}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 740 }(39, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)