from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([6,9,16]))
pari: [g,chi] = znchar(Mod(47,855))
Basic properties
Modulus: | \(855\) | |
Conductor: | \(855\) | 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 855.dh
\(\chi_{855}(47,\cdot)\) \(\chi_{855}(137,\cdot)\) \(\chi_{855}(218,\cdot)\) \(\chi_{855}(272,\cdot)\) \(\chi_{855}(302,\cdot)\) \(\chi_{855}(308,\cdot)\) \(\chi_{855}(347,\cdot)\) \(\chi_{855}(443,\cdot)\) \(\chi_{855}(473,\cdot)\) \(\chi_{855}(518,\cdot)\) \(\chi_{855}(662,\cdot)\) \(\chi_{855}(833,\cdot)\)
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: | 36.36.36045670002337036813834863966937246686386512405362460211785986211962997913360595703125.1 |
Values on generators
\((191,172,496)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{4}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 855 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
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)