from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,50,27]))
pari: [g,chi] = znchar(Mod(1787,1800))
Basic properties
| Modulus: | \(1800\) | |
| Conductor: | \(1800\) | 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 1800.dm
\(\chi_{1800}(83,\cdot)\) \(\chi_{1800}(203,\cdot)\) \(\chi_{1800}(227,\cdot)\) \(\chi_{1800}(347,\cdot)\) \(\chi_{1800}(563,\cdot)\) \(\chi_{1800}(587,\cdot)\) \(\chi_{1800}(803,\cdot)\) \(\chi_{1800}(923,\cdot)\) \(\chi_{1800}(947,\cdot)\) \(\chi_{1800}(1067,\cdot)\) \(\chi_{1800}(1163,\cdot)\) \(\chi_{1800}(1283,\cdot)\) \(\chi_{1800}(1427,\cdot)\) \(\chi_{1800}(1523,\cdot)\) \(\chi_{1800}(1667,\cdot)\) \(\chi_{1800}(1787,\cdot)\)
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
\((1351,901,1001,577)\) → \((-1,-1,e\left(\frac{5}{6}\right),e\left(\frac{9}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1800 }(1787, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) |
sage: chi.jacobi_sum(n)