from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(190, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([27,26]))
pari: [g,chi] = znchar(Mod(3,190))
Basic properties
Modulus: | \(190\) | |
Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{95}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 190.r
\(\chi_{190}(3,\cdot)\) \(\chi_{190}(13,\cdot)\) \(\chi_{190}(33,\cdot)\) \(\chi_{190}(53,\cdot)\) \(\chi_{190}(67,\cdot)\) \(\chi_{190}(97,\cdot)\) \(\chi_{190}(117,\cdot)\) \(\chi_{190}(127,\cdot)\) \(\chi_{190}(143,\cdot)\) \(\chi_{190}(147,\cdot)\) \(\chi_{190}(167,\cdot)\) \(\chi_{190}(173,\cdot)\)
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: | \(\Q(\zeta_{95})^+\) |
Values on generators
\((77,21)\) → \((-i,e\left(\frac{13}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 190 }(3, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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)