from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3240, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,49,27]))
pari: [g,chi] = znchar(Mod(119,3240))
Basic properties
| Modulus: | \(3240\) | |
| Conductor: | \(1620\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1620}(119,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3240.cy
\(\chi_{3240}(119,\cdot)\) \(\chi_{3240}(239,\cdot)\) \(\chi_{3240}(479,\cdot)\) \(\chi_{3240}(599,\cdot)\) \(\chi_{3240}(839,\cdot)\) \(\chi_{3240}(959,\cdot)\) \(\chi_{3240}(1199,\cdot)\) \(\chi_{3240}(1319,\cdot)\) \(\chi_{3240}(1559,\cdot)\) \(\chi_{3240}(1679,\cdot)\) \(\chi_{3240}(1919,\cdot)\) \(\chi_{3240}(2039,\cdot)\) \(\chi_{3240}(2279,\cdot)\) \(\chi_{3240}(2399,\cdot)\) \(\chi_{3240}(2639,\cdot)\) \(\chi_{3240}(2759,\cdot)\) \(\chi_{3240}(2999,\cdot)\) \(\chi_{3240}(3119,\cdot)\)
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
\((2431,1621,3161,1297)\) → \((-1,1,e\left(\frac{49}{54}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3240 }(119, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{54}\right)\) |
sage: chi.jacobi_sum(n)