from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2925, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,3,5]))
pari: [g,chi] = znchar(Mod(2,2925))
Basic properties
| Modulus: | \(2925\) | |
| Conductor: | \(2925\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2925.hr
\(\chi_{2925}(2,\cdot)\) \(\chi_{2925}(128,\cdot)\) \(\chi_{2925}(587,\cdot)\) \(\chi_{2925}(617,\cdot)\) \(\chi_{2925}(713,\cdot)\) \(\chi_{2925}(878,\cdot)\) \(\chi_{2925}(1172,\cdot)\) \(\chi_{2925}(1202,\cdot)\) \(\chi_{2925}(1298,\cdot)\) \(\chi_{2925}(1463,\cdot)\) \(\chi_{2925}(1787,\cdot)\) \(\chi_{2925}(1883,\cdot)\) \(\chi_{2925}(2048,\cdot)\) \(\chi_{2925}(2342,\cdot)\) \(\chi_{2925}(2372,\cdot)\) \(\chi_{2925}(2633,\cdot)\)
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
\((326,352,2251)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{20}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 2925 }(2, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) |
sage: chi.jacobi_sum(n)