from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2325, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,9,44]))
pari: [g,chi] = znchar(Mod(758,2325))
Basic properties
Modulus: | \(2325\) | |
Conductor: | \(2325\) | 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 2325.fp
\(\chi_{2325}(227,\cdot)\) \(\chi_{2325}(338,\cdot)\) \(\chi_{2325}(503,\cdot)\) \(\chi_{2325}(578,\cdot)\) \(\chi_{2325}(638,\cdot)\) \(\chi_{2325}(758,\cdot)\) \(\chi_{2325}(908,\cdot)\) \(\chi_{2325}(1073,\cdot)\) \(\chi_{2325}(1247,\cdot)\) \(\chi_{2325}(1322,\cdot)\) \(\chi_{2325}(1352,\cdot)\) \(\chi_{2325}(1967,\cdot)\) \(\chi_{2325}(2012,\cdot)\) \(\chi_{2325}(2117,\cdot)\) \(\chi_{2325}(2273,\cdot)\) \(\chi_{2325}(2312,\cdot)\)
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
\((776,652,1801)\) → \((-1,e\left(\frac{3}{20}\right),e\left(\frac{11}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 2325 }(758, a) \) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{17}{60}\right)\) | \(-i\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{30}\right)\) |
sage: chi.jacobi_sum(n)