from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3960, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,50,15,54]))
pari: [g,chi] = znchar(Mod(3317,3960))
Basic properties
| Modulus: | \(3960\) | |
| Conductor: | \(3960\) | 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 3960.hj
\(\chi_{3960}(173,\cdot)\) \(\chi_{3960}(293,\cdot)\) \(\chi_{3960}(437,\cdot)\) \(\chi_{3960}(677,\cdot)\) \(\chi_{3960}(893,\cdot)\) \(\chi_{3960}(1157,\cdot)\) \(\chi_{3960}(1613,\cdot)\) \(\chi_{3960}(1733,\cdot)\) \(\chi_{3960}(1757,\cdot)\) \(\chi_{3960}(1877,\cdot)\) \(\chi_{3960}(2477,\cdot)\) \(\chi_{3960}(2813,\cdot)\) \(\chi_{3960}(3053,\cdot)\) \(\chi_{3960}(3197,\cdot)\) \(\chi_{3960}(3317,\cdot)\) \(\chi_{3960}(3533,\cdot)\)
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
\((991,1981,3521,2377,2521)\) → \((1,-1,e\left(\frac{5}{6}\right),i,e\left(\frac{9}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 3960 }(3317, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)