from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(864, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,32]))
pari: [g,chi] = znchar(Mod(7,864))
Basic properties
| Modulus: | \(864\) | |
| Conductor: | \(432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{432}(331,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 864.bp
\(\chi_{864}(7,\cdot)\) \(\chi_{864}(103,\cdot)\) \(\chi_{864}(151,\cdot)\) \(\chi_{864}(247,\cdot)\) \(\chi_{864}(295,\cdot)\) \(\chi_{864}(391,\cdot)\) \(\chi_{864}(439,\cdot)\) \(\chi_{864}(535,\cdot)\) \(\chi_{864}(583,\cdot)\) \(\chi_{864}(679,\cdot)\) \(\chi_{864}(727,\cdot)\) \(\chi_{864}(823,\cdot)\)
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: | 36.0.614667125325361522818798575155151578949632894783197825857500612833312768.1 |
Values on generators
\((703,325,353)\) → \((-1,i,e\left(\frac{8}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 864 }(7, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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)