from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85600, base_ring=CyclotomicField(2120))
M = H._module
chi = DirichletCharacter(H, M([1060,1325,212,240]))
pari: [g,chi] = znchar(Mod(779,85600))
Basic properties
| Modulus: | \(85600\) | |
| Conductor: | \(85600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2120\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 85600.ii
\(\chi_{85600}(19,\cdot)\) \(\chi_{85600}(539,\cdot)\) \(\chi_{85600}(579,\cdot)\) \(\chi_{85600}(779,\cdot)\) \(\chi_{85600}(859,\cdot)\) \(\chi_{85600}(939,\cdot)\) \(\chi_{85600}(979,\cdot)\) \(\chi_{85600}(1019,\cdot)\) \(\chi_{85600}(1139,\cdot)\) \(\chi_{85600}(1219,\cdot)\) \(\chi_{85600}(1539,\cdot)\)
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: | $\Q(\zeta_{2120})$ |
| Fixed field: | Number field defined by a degree 2120 polynomial (not computed) |
Values on generators
\((26751,32101,82177,16801)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{10}\right),e\left(\frac{6}{53}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 85600 }(779, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2119}{2120}\right)\) | \(e\left(\frac{25}{212}\right)\) | \(e\left(\frac{1059}{1060}\right)\) | \(e\left(\frac{1517}{2120}\right)\) | \(e\left(\frac{1823}{2120}\right)\) | \(e\left(\frac{22}{265}\right)\) | \(e\left(\frac{1071}{2120}\right)\) | \(e\left(\frac{249}{2120}\right)\) | \(e\left(\frac{391}{1060}\right)\) | \(e\left(\frac{2117}{2120}\right)\) |
sage: chi.jacobi_sum(n)