from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,27,25]))
pari: [g,chi] = znchar(Mod(719,4725))
Basic properties
Modulus: | \(4725\) | |
Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1575}(194,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.dx
\(\chi_{4725}(719,\cdot)\) \(\chi_{4725}(1664,\cdot)\) \(\chi_{4725}(1844,\cdot)\) \(\chi_{4725}(2609,\cdot)\) \(\chi_{4725}(2789,\cdot)\) \(\chi_{4725}(3554,\cdot)\) \(\chi_{4725}(3734,\cdot)\) \(\chi_{4725}(4679,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((4376,1702,2026)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 4725 }(719, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) |
sage: chi.jacobi_sum(n)