from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4950, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,9,24]))
pari: [g,chi] = znchar(Mod(533,4950))
Basic properties
| Modulus: | \(4950\) | |
| Conductor: | \(2475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2475}(533,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4950.gh
\(\chi_{4950}(533,\cdot)\) \(\chi_{4950}(587,\cdot)\) \(\chi_{4950}(1703,\cdot)\) \(\chi_{4950}(2027,\cdot)\) \(\chi_{4950}(2117,\cdot)\) \(\chi_{4950}(2183,\cdot)\) \(\chi_{4950}(2237,\cdot)\) \(\chi_{4950}(2297,\cdot)\) \(\chi_{4950}(2423,\cdot)\) \(\chi_{4950}(2963,\cdot)\) \(\chi_{4950}(3353,\cdot)\) \(\chi_{4950}(3677,\cdot)\) \(\chi_{4950}(3767,\cdot)\) \(\chi_{4950}(3947,\cdot)\) \(\chi_{4950}(4073,\cdot)\) \(\chi_{4950}(4613,\cdot)\)
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
\((551,2377,4501)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4950 }(533, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)