from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2166, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,4]))
pari: [g,chi] = znchar(Mod(115,2166))
Basic properties
| Modulus: | \(2166\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{361}(115,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2166.m
\(\chi_{2166}(115,\cdot)\) \(\chi_{2166}(229,\cdot)\) \(\chi_{2166}(343,\cdot)\) \(\chi_{2166}(457,\cdot)\) \(\chi_{2166}(571,\cdot)\) \(\chi_{2166}(685,\cdot)\) \(\chi_{2166}(799,\cdot)\) \(\chi_{2166}(913,\cdot)\) \(\chi_{2166}(1027,\cdot)\) \(\chi_{2166}(1141,\cdot)\) \(\chi_{2166}(1255,\cdot)\) \(\chi_{2166}(1369,\cdot)\) \(\chi_{2166}(1483,\cdot)\) \(\chi_{2166}(1597,\cdot)\) \(\chi_{2166}(1711,\cdot)\) \(\chi_{2166}(1825,\cdot)\) \(\chi_{2166}(1939,\cdot)\) \(\chi_{2166}(2053,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 19.19.10842505080063916320800450434338728415281531281.1 |
Values on generators
\((1445,1807)\) → \((1,e\left(\frac{2}{19}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2166 }(115, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) |
sage: chi.jacobi_sum(n)