from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,9,50]))
pari: [g,chi] = znchar(Mod(208,4725))
Basic properties
Modulus: | \(4725\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1575}(1258,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.fo
\(\chi_{4725}(208,\cdot)\) \(\chi_{4725}(388,\cdot)\) \(\chi_{4725}(397,\cdot)\) \(\chi_{4725}(577,\cdot)\) \(\chi_{4725}(1153,\cdot)\) \(\chi_{4725}(1333,\cdot)\) \(\chi_{4725}(1342,\cdot)\) \(\chi_{4725}(1522,\cdot)\) \(\chi_{4725}(2098,\cdot)\) \(\chi_{4725}(2278,\cdot)\) \(\chi_{4725}(2287,\cdot)\) \(\chi_{4725}(2467,\cdot)\) \(\chi_{4725}(3223,\cdot)\) \(\chi_{4725}(3412,\cdot)\) \(\chi_{4725}(3988,\cdot)\) \(\chi_{4725}(4177,\cdot)\)
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
\((4376,1702,2026)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 4725 }(208, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)