from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4620, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,15,20,18]))
pari: [g,chi] = znchar(Mod(107,4620))
Basic properties
| Modulus: | \(4620\) | |
| Conductor: | \(4620\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4620.gy
\(\chi_{4620}(107,\cdot)\) \(\chi_{4620}(347,\cdot)\) \(\chi_{4620}(767,\cdot)\) \(\chi_{4620}(1283,\cdot)\) \(\chi_{4620}(1943,\cdot)\) \(\chi_{4620}(2207,\cdot)\) \(\chi_{4620}(2543,\cdot)\) \(\chi_{4620}(2867,\cdot)\) \(\chi_{4620}(3203,\cdot)\) \(\chi_{4620}(3383,\cdot)\) \(\chi_{4620}(3467,\cdot)\) \(\chi_{4620}(3803,\cdot)\) \(\chi_{4620}(4043,\cdot)\) \(\chi_{4620}(4127,\cdot)\) \(\chi_{4620}(4307,\cdot)\) \(\chi_{4620}(4463,\cdot)\)
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
\((2311,1541,3697,661,2521)\) → \((-1,-1,i,e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 4620 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{19}{60}\right)\) |
sage: chi.jacobi_sum(n)