from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,24,35]))
pari: [g,chi] = znchar(Mod(258,9295))
Basic properties
| Modulus: | \(9295\) | |
| Conductor: | \(715\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{715}(258,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9295.dv
\(\chi_{9295}(258,\cdot)\) \(\chi_{9295}(357,\cdot)\) \(\chi_{9295}(488,\cdot)\) \(\chi_{9295}(587,\cdot)\) \(\chi_{9295}(1103,\cdot)\) \(\chi_{9295}(1202,\cdot)\) \(\chi_{9295}(3023,\cdot)\) \(\chi_{9295}(3122,\cdot)\) \(\chi_{9295}(4713,\cdot)\) \(\chi_{9295}(4812,\cdot)\) \(\chi_{9295}(5328,\cdot)\) \(\chi_{9295}(5427,\cdot)\) \(\chi_{9295}(5558,\cdot)\) \(\chi_{9295}(5657,\cdot)\) \(\chi_{9295}(7863,\cdot)\) \(\chi_{9295}(7962,\cdot)\)
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
\((7437,4226,6931)\) → \((-i,e\left(\frac{2}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 9295 }(258, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)