from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,20,21]))
pari: [g,chi] = znchar(Mod(103,3600))
Basic properties
| Modulus: | \(3600\) | |
| Conductor: | \(1800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1800}(1003,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.fz
\(\chi_{3600}(103,\cdot)\) \(\chi_{3600}(247,\cdot)\) \(\chi_{3600}(583,\cdot)\) \(\chi_{3600}(727,\cdot)\) \(\chi_{3600}(823,\cdot)\) \(\chi_{3600}(967,\cdot)\) \(\chi_{3600}(1303,\cdot)\) \(\chi_{3600}(1447,\cdot)\) \(\chi_{3600}(1687,\cdot)\) \(\chi_{3600}(2023,\cdot)\) \(\chi_{3600}(2167,\cdot)\) \(\chi_{3600}(2263,\cdot)\) \(\chi_{3600}(2887,\cdot)\) \(\chi_{3600}(2983,\cdot)\) \(\chi_{3600}(3127,\cdot)\) \(\chi_{3600}(3463,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((3151,901,2801,577)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)