sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(2850, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([9,9,7]))
pari: [g,chi] = znchar(Mod(299,2850))
Basic properties
| Modulus: | \(2850\) | |
| Conductor: | \(285\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{285}(14,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2850.bo
\(\chi_{2850}(299,\cdot)\) \(\chi_{2850}(599,\cdot)\) \(\chi_{2850}(1199,\cdot)\) \(\chi_{2850}(1649,\cdot)\) \(\chi_{2850}(1799,\cdot)\) \(\chi_{2850}(2549,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | 18.18.210684481487848166847338548828125.1 |
Values on generators
\((1901,1027,1351)\) → \((-1,-1,e\left(\frac{7}{18}\right))\)
Values
| \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) |