from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,9,0]))
pari: [g,chi] = znchar(Mod(1958,4275))
Basic properties
Modulus: | \(4275\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(158,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.fr
\(\chi_{4275}(77,\cdot)\) \(\chi_{4275}(248,\cdot)\) \(\chi_{4275}(362,\cdot)\) \(\chi_{4275}(533,\cdot)\) \(\chi_{4275}(1103,\cdot)\) \(\chi_{4275}(1217,\cdot)\) \(\chi_{4275}(1388,\cdot)\) \(\chi_{4275}(1787,\cdot)\) \(\chi_{4275}(1958,\cdot)\) \(\chi_{4275}(2072,\cdot)\) \(\chi_{4275}(2642,\cdot)\) \(\chi_{4275}(2813,\cdot)\) \(\chi_{4275}(2927,\cdot)\) \(\chi_{4275}(3098,\cdot)\) \(\chi_{4275}(3497,\cdot)\) \(\chi_{4275}(3953,\cdot)\)
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
\((1901,1027,1351)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 4275 }(1958, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) |
sage: chi.jacobi_sum(n)