from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,41,27]))
pari: [g,chi] = znchar(Mod(59,1620))
Basic properties
| Modulus: | \(1620\) | |
| Conductor: | \(1620\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | 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 1620.bm
\(\chi_{1620}(59,\cdot)\) \(\chi_{1620}(119,\cdot)\) \(\chi_{1620}(239,\cdot)\) \(\chi_{1620}(299,\cdot)\) \(\chi_{1620}(419,\cdot)\) \(\chi_{1620}(479,\cdot)\) \(\chi_{1620}(599,\cdot)\) \(\chi_{1620}(659,\cdot)\) \(\chi_{1620}(779,\cdot)\) \(\chi_{1620}(839,\cdot)\) \(\chi_{1620}(959,\cdot)\) \(\chi_{1620}(1019,\cdot)\) \(\chi_{1620}(1139,\cdot)\) \(\chi_{1620}(1199,\cdot)\) \(\chi_{1620}(1319,\cdot)\) \(\chi_{1620}(1379,\cdot)\) \(\chi_{1620}(1499,\cdot)\) \(\chi_{1620}(1559,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((811,1541,1297)\) → \((-1,e\left(\frac{41}{54}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1620 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{54}\right)\) |
sage: chi.jacobi_sum(n)