from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(751, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([38]))
pari: [g,chi] = znchar(Mod(193,751))
Basic properties
| Modulus: | \(751\) | |
| Conductor: | \(751\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | 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 751.h
\(\chi_{751}(51,\cdot)\) \(\chi_{751}(53,\cdot)\) \(\chi_{751}(117,\cdot)\) \(\chi_{751}(162,\cdot)\) \(\chi_{751}(171,\cdot)\) \(\chi_{751}(179,\cdot)\) \(\chi_{751}(193,\cdot)\) \(\chi_{751}(325,\cdot)\) \(\chi_{751}(348,\cdot)\) \(\chi_{751}(420,\cdot)\) \(\chi_{751}(450,\cdot)\) \(\chi_{751}(466,\cdot)\) \(\chi_{751}(475,\cdot)\) \(\chi_{751}(481,\cdot)\) \(\chi_{751}(485,\cdot)\) \(\chi_{751}(499,\cdot)\) \(\chi_{751}(556,\cdot)\) \(\chi_{751}(666,\cdot)\) \(\chi_{751}(703,\cdot)\) \(\chi_{751}(710,\cdot)\)
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 25 polynomial |
Values on generators
\(3\) → \(e\left(\frac{19}{25}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 751 }(193, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) |
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)