from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4950, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,15,42]))
pari: [g,chi] = znchar(Mod(7,4950))
Basic properties
| Modulus: | \(4950\) | |
| Conductor: | \(495\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{495}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4950.fw
\(\chi_{4950}(7,\cdot)\) \(\chi_{4950}(193,\cdot)\) \(\chi_{4950}(457,\cdot)\) \(\chi_{4950}(607,\cdot)\) \(\chi_{4950}(943,\cdot)\) \(\chi_{4950}(1393,\cdot)\) \(\chi_{4950}(1843,\cdot)\) \(\chi_{4950}(1993,\cdot)\) \(\chi_{4950}(2257,\cdot)\) \(\chi_{4950}(2857,\cdot)\) \(\chi_{4950}(3307,\cdot)\) \(\chi_{4950}(3643,\cdot)\) \(\chi_{4950}(3757,\cdot)\) \(\chi_{4950}(4243,\cdot)\) \(\chi_{4950}(4507,\cdot)\) \(\chi_{4950}(4693,\cdot)\)
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
\((551,2377,4501)\) → \((e\left(\frac{2}{3}\right),i,e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4950 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)