from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85600, base_ring=CyclotomicField(2120))
M = H._module
chi = DirichletCharacter(H, M([0,795,1378,1100]))
pari: [g,chi] = znchar(Mod(317,85600))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 85600.io
\(\chi_{85600}(77,\cdot)\) \(\chi_{85600}(133,\cdot)\) \(\chi_{85600}(317,\cdot)\) \(\chi_{85600}(613,\cdot)\) \(\chi_{85600}(773,\cdot)\) \(\chi_{85600}(853,\cdot)\) \(\chi_{85600}(877,\cdot)\) \(\chi_{85600}(933,\cdot)\) \(\chi_{85600}(1013,\cdot)\) \(\chi_{85600}(1037,\cdot)\) \(\chi_{85600}(1173,\cdot)\) \(\chi_{85600}(1197,\cdot)\) \(\chi_{85600}(1413,\cdot)\) \(\chi_{85600}(1437,\cdot)\)
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
\((26751,32101,82177,16801)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{13}{20}\right),e\left(\frac{55}{106}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 85600 }(317, a) \) | \(1\) | \(1\) | \(e\left(\frac{2111}{2120}\right)\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{1051}{1060}\right)\) | \(e\left(\frac{1463}{2120}\right)\) | \(e\left(\frac{507}{2120}\right)\) | \(e\left(\frac{1057}{1060}\right)\) | \(e\left(\frac{1689}{2120}\right)\) | \(e\left(\frac{651}{2120}\right)\) | \(e\left(\frac{151}{265}\right)\) | \(e\left(\frac{2093}{2120}\right)\) |
sage: chi.jacobi_sum(n)