from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(755, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,34]))
pari: [g,chi] = znchar(Mod(624,755))
Basic properties
| Modulus: | \(755\) | |
| Conductor: | \(755\) | 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 755.z
\(\chi_{755}(9,\cdot)\) \(\chi_{755}(29,\cdot)\) \(\chi_{755}(44,\cdot)\) \(\chi_{755}(84,\cdot)\) \(\chi_{755}(94,\cdot)\) \(\chi_{755}(124,\cdot)\) \(\chi_{755}(219,\cdot)\) \(\chi_{755}(229,\cdot)\) \(\chi_{755}(249,\cdot)\) \(\chi_{755}(274,\cdot)\) \(\chi_{755}(299,\cdot)\) \(\chi_{755}(374,\cdot)\) \(\chi_{755}(429,\cdot)\) \(\chi_{755}(534,\cdot)\) \(\chi_{755}(539,\cdot)\) \(\chi_{755}(544,\cdot)\) \(\chi_{755}(624,\cdot)\) \(\chi_{755}(654,\cdot)\) \(\chi_{755}(714,\cdot)\) \(\chi_{755}(729,\cdot)\)
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
\((152,6)\) → \((-1,e\left(\frac{17}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 755 }(624, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{50}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{19}{50}\right)\) |
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)