from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,10,9]))
pari: [g,chi] = znchar(Mod(83,3600))
Basic properties
| Modulus: | \(3600\) | |
| Conductor: | \(3600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.fp
\(\chi_{3600}(83,\cdot)\) \(\chi_{3600}(347,\cdot)\) \(\chi_{3600}(563,\cdot)\) \(\chi_{3600}(587,\cdot)\) \(\chi_{3600}(803,\cdot)\) \(\chi_{3600}(1067,\cdot)\) \(\chi_{3600}(1283,\cdot)\) \(\chi_{3600}(1523,\cdot)\) \(\chi_{3600}(1787,\cdot)\) \(\chi_{3600}(2003,\cdot)\) \(\chi_{3600}(2027,\cdot)\) \(\chi_{3600}(2723,\cdot)\) \(\chi_{3600}(2747,\cdot)\) \(\chi_{3600}(2963,\cdot)\) \(\chi_{3600}(3227,\cdot)\) \(\chi_{3600}(3467,\cdot)\)
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
\((3151,901,2801,577)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)