from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3020, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,52]))
pari: [g,chi] = znchar(Mod(2583,3020))
Basic properties
| Modulus: | \(3020\) | |
| Conductor: | \(3020\) | 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 3020.ca
\(\chi_{3020}(167,\cdot)\) \(\chi_{3020}(227,\cdot)\) \(\chi_{3020}(387,\cdot)\) \(\chi_{3020}(407,\cdot)\) \(\chi_{3020}(883,\cdot)\) \(\chi_{3020}(1363,\cdot)\) \(\chi_{3020}(1487,\cdot)\) \(\chi_{3020}(1663,\cdot)\) \(\chi_{3020}(1967,\cdot)\) \(\chi_{3020}(2267,\cdot)\) \(\chi_{3020}(2303,\cdot)\) \(\chi_{3020}(2583,\cdot)\) \(\chi_{3020}(2643,\cdot)\) \(\chi_{3020}(2803,\cdot)\) \(\chi_{3020}(2823,\cdot)\) \(\chi_{3020}(2907,\cdot)\)
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
\((1511,2417,761)\) → \((-1,-i,e\left(\frac{13}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3020 }(2583, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{20}\right)\) |
sage: chi.jacobi_sum(n)