from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,33,20]))
pari: [g,chi] = znchar(Mod(548,4725))
Basic properties
| Modulus: | \(4725\) | |
| Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1575}(1073,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.fx
\(\chi_{4725}(548,\cdot)\) \(\chi_{4725}(737,\cdot)\) \(\chi_{4725}(1313,\cdot)\) \(\chi_{4725}(1502,\cdot)\) \(\chi_{4725}(2258,\cdot)\) \(\chi_{4725}(2438,\cdot)\) \(\chi_{4725}(2447,\cdot)\) \(\chi_{4725}(2627,\cdot)\) \(\chi_{4725}(3203,\cdot)\) \(\chi_{4725}(3383,\cdot)\) \(\chi_{4725}(3392,\cdot)\) \(\chi_{4725}(3572,\cdot)\) \(\chi_{4725}(4148,\cdot)\) \(\chi_{4725}(4328,\cdot)\) \(\chi_{4725}(4337,\cdot)\) \(\chi_{4725}(4517,\cdot)\)
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
\((4376,1702,2026)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{11}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 4725 }(548, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)