from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2475, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,9,36]))
pari: [g,chi] = znchar(Mod(1208,2475))
Basic properties
Modulus: | \(2475\) | |
Conductor: | \(2475\) | 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 2475.gk
\(\chi_{2475}(113,\cdot)\) \(\chi_{2475}(203,\cdot)\) \(\chi_{2475}(212,\cdot)\) \(\chi_{2475}(317,\cdot)\) \(\chi_{2475}(383,\cdot)\) \(\chi_{2475}(752,\cdot)\) \(\chi_{2475}(797,\cdot)\) \(\chi_{2475}(938,\cdot)\) \(\chi_{2475}(1028,\cdot)\) \(\chi_{2475}(1037,\cdot)\) \(\chi_{2475}(1148,\cdot)\) \(\chi_{2475}(1208,\cdot)\) \(\chi_{2475}(1577,\cdot)\) \(\chi_{2475}(1622,\cdot)\) \(\chi_{2475}(1967,\cdot)\) \(\chi_{2475}(1973,\cdot)\)
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
\((551,2377,2026)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right),e\left(\frac{3}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 2475 }(1208, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(-i\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-1\) | \(e\left(\frac{29}{60}\right)\) |
sage: chi.jacobi_sum(n)