from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(125, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([47]))
pari: [g,chi] = znchar(Mod(84,125))
Basic properties
| Modulus: | \(125\) | |
| Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 125.h
\(\chi_{125}(4,\cdot)\) \(\chi_{125}(9,\cdot)\) \(\chi_{125}(14,\cdot)\) \(\chi_{125}(19,\cdot)\) \(\chi_{125}(29,\cdot)\) \(\chi_{125}(34,\cdot)\) \(\chi_{125}(39,\cdot)\) \(\chi_{125}(44,\cdot)\) \(\chi_{125}(54,\cdot)\) \(\chi_{125}(59,\cdot)\) \(\chi_{125}(64,\cdot)\) \(\chi_{125}(69,\cdot)\) \(\chi_{125}(79,\cdot)\) \(\chi_{125}(84,\cdot)\) \(\chi_{125}(89,\cdot)\) \(\chi_{125}(94,\cdot)\) \(\chi_{125}(104,\cdot)\) \(\chi_{125}(109,\cdot)\) \(\chi_{125}(114,\cdot)\) \(\chi_{125}(119,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\(2\) → \(e\left(\frac{47}{50}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 125 }(84, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{33}{50}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)