from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3850, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([57,40,30]))
pari: [g,chi] = znchar(Mod(263,3850))
Basic properties
| Modulus: | \(3850\) | |
| Conductor: | \(1925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1925}(263,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3850.gb
\(\chi_{3850}(263,\cdot)\) \(\chi_{3850}(373,\cdot)\) \(\chi_{3850}(417,\cdot)\) \(\chi_{3850}(527,\cdot)\) \(\chi_{3850}(1033,\cdot)\) \(\chi_{3850}(1187,\cdot)\) \(\chi_{3850}(1297,\cdot)\) \(\chi_{3850}(1803,\cdot)\) \(\chi_{3850}(1913,\cdot)\) \(\chi_{3850}(2067,\cdot)\) \(\chi_{3850}(2573,\cdot)\) \(\chi_{3850}(2683,\cdot)\) \(\chi_{3850}(2727,\cdot)\) \(\chi_{3850}(2837,\cdot)\) \(\chi_{3850}(3453,\cdot)\) \(\chi_{3850}(3497,\cdot)\)
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
\((2927,2201,1751)\) → \((e\left(\frac{19}{20}\right),e\left(\frac{2}{3}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3850 }(263, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) |
sage: chi.jacobi_sum(n)