from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87362, base_ring=CyclotomicField(3762))
M = H._module
chi = DirichletCharacter(H, M([2394,1606]))
pari: [g,chi] = znchar(Mod(23,87362))
Basic properties
| Modulus: | \(87362\) | |
| Conductor: | \(43681\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1881\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{43681}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 87362.dd
\(\chi_{87362}(23,\cdot)\) \(\chi_{87362}(111,\cdot)\) \(\chi_{87362}(177,\cdot)\) \(\chi_{87362}(199,\cdot)\) \(\chi_{87362}(309,\cdot)\) \(\chi_{87362}(397,\cdot)\) \(\chi_{87362}(441,\cdot)\) \(\chi_{87362}(529,\cdot)\) \(\chi_{87362}(617,\cdot)\) \(\chi_{87362}(815,\cdot)\) \(\chi_{87362}(859,\cdot)\) \(\chi_{87362}(947,\cdot)\) \(\chi_{87362}(1013,\cdot)\) \(\chi_{87362}(1035,\cdot)\) \(\chi_{87362}(1233,\cdot)\) \(\chi_{87362}(1277,\cdot)\) \(\chi_{87362}(1365,\cdot)\) \(\chi_{87362}(1431,\cdot)\)
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: | $\Q(\zeta_{1881})$ |
| Fixed field: | Number field defined by a degree 1881 polynomial (not computed) |
Values on generators
\((21661,22023)\) → \((e\left(\frac{7}{11}\right),e\left(\frac{73}{171}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 87362 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{248}{1881}\right)\) | \(e\left(\frac{307}{627}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{1360}{1881}\right)\) | \(e\left(\frac{886}{1881}\right)\) | \(e\left(\frac{1244}{1881}\right)\) | \(e\left(\frac{1559}{1881}\right)\) | \(e\left(\frac{1642}{1881}\right)\) | \(e\left(\frac{496}{1881}\right)\) |
sage: chi.jacobi_sum(n)