from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85600, base_ring=CyclotomicField(2120))
M = H._module
chi = DirichletCharacter(H, M([0,1325,1272,140]))
pari: [g,chi] = znchar(Mod(21,85600))
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.ij
\(\chi_{85600}(21,\cdot)\) \(\chi_{85600}(181,\cdot)\) \(\chi_{85600}(221,\cdot)\) \(\chi_{85600}(341,\cdot)\) \(\chi_{85600}(381,\cdot)\) \(\chi_{85600}(541,\cdot)\) \(\chi_{85600}(581,\cdot)\) \(\chi_{85600}(781,\cdot)\) \(\chi_{85600}(821,\cdot)\) \(\chi_{85600}(861,\cdot)\) \(\chi_{85600}(981,\cdot)\) \(\chi_{85600}(1021,\cdot)\) \(\chi_{85600}(1061,\cdot)\) \(\chi_{85600}(1141,\cdot)\) \(\chi_{85600}(1261,\cdot)\) \(\chi_{85600}(1381,\cdot)\) \(\chi_{85600}(1461,\cdot)\) \(\chi_{85600}(1541,\cdot)\)
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
\((26751,32101,82177,16801)\) → \((1,e\left(\frac{5}{8}\right),e\left(\frac{3}{5}\right),e\left(\frac{7}{106}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 85600 }(21, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1479}{2120}\right)\) | \(e\left(\frac{19}{212}\right)\) | \(e\left(\frac{419}{1060}\right)\) | \(e\left(\frac{377}{2120}\right)\) | \(e\left(\frac{1483}{2120}\right)\) | \(e\left(\frac{57}{265}\right)\) | \(e\left(\frac{691}{2120}\right)\) | \(e\left(\frac{1669}{2120}\right)\) | \(e\left(\frac{471}{1060}\right)\) | \(e\left(\frac{197}{2120}\right)\) |
sage: chi.jacobi_sum(n)