from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3895, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,10,6]))
pari: [g,chi] = znchar(Mod(312,3895))
Basic properties
| Modulus: | \(3895\) | |
| Conductor: | \(3895\) | 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 3895.eo
\(\chi_{3895}(107,\cdot)\) \(\chi_{3895}(312,\cdot)\) \(\chi_{3895}(373,\cdot)\) \(\chi_{3895}(392,\cdot)\) \(\chi_{3895}(578,\cdot)\) \(\chi_{3895}(597,\cdot)\) \(\chi_{3895}(1152,\cdot)\) \(\chi_{3895}(1357,\cdot)\) \(\chi_{3895}(2368,\cdot)\) \(\chi_{3895}(2573,\cdot)\) \(\chi_{3895}(3147,\cdot)\) \(\chi_{3895}(3223,\cdot)\) \(\chi_{3895}(3352,\cdot)\) \(\chi_{3895}(3428,\cdot)\) \(\chi_{3895}(3508,\cdot)\) \(\chi_{3895}(3713,\cdot)\)
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
\((3117,2871,1236)\) → \((i,e\left(\frac{1}{6}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 3895 }(312, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) |
sage: chi.jacobi_sum(n)