from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4950, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,0,3]))
pari: [g,chi] = znchar(Mod(701,4950))
Basic properties
| Modulus: | \(4950\) | |
| Conductor: | \(33\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{33}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4950.cc
\(\chi_{4950}(701,\cdot)\) \(\chi_{4950}(1151,\cdot)\) \(\chi_{4950}(1601,\cdot)\) \(\chi_{4950}(3401,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | \(\Q(\zeta_{33})^+\) |
Values on generators
\((551,2377,4501)\) → \((-1,1,e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 4950 }(701, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)