from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,39,0]))
pari: [g,chi] = znchar(Mod(67,2475))
Basic properties
Modulus: | \(2475\) | |
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 \(\chi_{225}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2475.fx
\(\chi_{2475}(67,\cdot)\) \(\chi_{2475}(133,\cdot)\) \(\chi_{2475}(463,\cdot)\) \(\chi_{2475}(562,\cdot)\) \(\chi_{2475}(628,\cdot)\) \(\chi_{2475}(727,\cdot)\) \(\chi_{2475}(958,\cdot)\) \(\chi_{2475}(1123,\cdot)\) \(\chi_{2475}(1222,\cdot)\) \(\chi_{2475}(1453,\cdot)\) \(\chi_{2475}(1552,\cdot)\) \(\chi_{2475}(1717,\cdot)\) \(\chi_{2475}(1948,\cdot)\) \(\chi_{2475}(2047,\cdot)\) \(\chi_{2475}(2113,\cdot)\) \(\chi_{2475}(2212,\cdot)\)
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
\((551,2377,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{13}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 2475 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) |
sage: chi.jacobi_sum(n)